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Inverse Dynamic Analysis of the Human Locomotion Apparatus for Gait
Helder Jorge Carrapatoso Oliveira
Thesis to obtain the Master of Science Degree in Mechanical Engineering
Supervisors: Prof. Carlos Miguel Fernandes Quental Prof. João Orlando Marques Gameiro Folgado
Examination Committee
Chairperson: Prof. Rui Manuel dos Santos Oliveira Baptista Supervisor: Prof. Carlos Miguel Fernandes Quental Member of the committee: Prof. Miguel Pedro Tavares da Silva
November 2016
Resumo
Neste trabalho é apresentada uma metodologia para realizar uma análise dinâmica inversa da marcha humana. No contexto de dinâmica multicorpo apresenta-se, um modelo biomecânico tridimensional utilizando coordenadas cartesianas. O modelo possui quarenta e um graus de liberdade e é constituído por oito juntas esféricas, quatro juntas universais e três juntas de revolução. Para executar a análise de dinâmica inversa é necessário adquirir os dados cinemáticos e cinéticos da marcha do sujeito de estudo bem como estimar as propriedades geométricas e físicas de cada segmento anatómico. Considerando actuadores rotacionais de guiamento nas juntas para cada um dos graus de liberdade do modelo, a solução da análise de dinâmica inversa é determinada através da resolução das equações do movimento, permitindo a obtenção das forças internas geradas pelos corpos e os momentos intersegmentares. Estes resultados demonstram conformidade com os obtidos na literatura. O método clássico de diagrama de corpo livre para o equilíbrio de forças é utilizado para validar de forma independente os resultados da formulação multicorpo. As forças musculares desenvolvidas pelo aparelho locomotor são avaliadas substituindo os actuadores rotacionais de guiamento das juntas por actuadores musculares, descritos pelo modelo muscular do tipo Hill. Este sistema muscular é redundante devido ao número de variáveis desconhecidas ser superior ao número de equações do movimento disponíveis. O problema indeterminado é resolvido através de um processo de optimização, no qual se procura a solução que minimiza um determinado critério com relevância biológica. Duas funções objectivo, baseadas em critérios fisiológicos, são aplicadas e os seus resultados discutidos.
Palavras-Chave
Biomecânica Análise de marcha Dinâmica multicorpo Dinâmica inversa Forças musculares redundantes Optimização
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Abstract
A methodology for the inverse dynamic analysis of human gait is presented in this work. In the context of multibody dynamics, a three-dimensional biomechanical model is presented using Cartesian coordinates. This model has forty-one degrees of freedom and is constituted by eight spherical joints, four universal joints and three revolute joints, which are controlled by rotational driver actuators. To perform an inverse dynamic analysis is required to acquire the gait of the subject under analysis as well as his geometrical and physical properties of each anatomical segment. Considering rotational driver actuators at the joints for each degree of freedom of the model, the solution of the inverse dynamic analysis is determined by solving the equations of motion, allowing to obtain the internal forces generated by the bodies and joint torques developed by the actuators. These results show agreement with those shown in the literature. The classical method free body diagram for the equilibrium of forces is used to validate the multibody formulation results. The muscle forces developed by the lower extremity apparatus are evaluated substituting the rotational driver actuators of the joints by muscle actuators, described by the Hill type muscle model. This muscular system is redundant, since the number of unknowns is higher than the number of available equations of motion. The indeterminate problem is solved by an optimization process, in which the solution that minimizes specific criteria with biological relevance is searched. Two minimization objective functions, based on physiological criteria, are applied and their results discussed.
Keywords
Biomechanics Gait analysis Multibody dynamics Inverse dynamics Redundant muscle forces Optimization
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Acknowledgments
Ao Professor Carlos Quental, quero agradecer todo o apoio, coordenação e empenho no desenvolvimento deste trabalho. A convivência quase diária ajudou na compreensão e resolução dos problemas que foram surgindo. Para sempre ficará uma enorme amizade. Ao Professor João Folgado, quero agradecer todo o apoio e motivação no desenvolvimento deste trabalho. Ao Professor Jorge Ambrósio, quero agradecer por me ter cativado na disciplina de Dinâmica de Sistemas Mecânicos e pela oportunidade de trabalhar na área da Biomecânica. Embora não sendo meu orientador científico, esteve sempre presente em todas as etapas deste trabalho. Obrigado pelas imersivas aulas, reuniões com os colegas de curso e pela partilha de conhecimento. Terá sempre o meu respeito e admiração. Ao Sérgio Gonçalves, quero agradecer a ajuda na aquisição dos ensaios experimentais, troca de ideias e esclarecimento de dúvidas a qualquer hora do dia. Obrigado pela paciência e amizade. To Stefanie Brändle, my office colleague and friend, I would like to thank the share of ideas and support in the development of this work. I will not forget your friendly smile even in the worst problems we both faced. Also want to thank Hugo Magalhães for being our organizer of various activities we shared with our friends Angela Bautista and Rafael Cordeiro. To Professors Carol Teitz and Dan Graney of the University of Washington I would like to thank the authorization to use in this dissertation the images of the lower extremity muscles extracted from the site ‘Musculoskeletal Atlas: A Musculoskeletal Atlas of the Human Body’. A todos os meus colegas e amigos que me apoiaram e motivaram ao longo do curso e no desenvolvimento deste trabalho. Destaco o apoio e amizade do André Brás ao longo de todo o percurso académico. Um muito obrigado ao Ricardo Torres, que embora estando longe, sempre transmitiu uma motivação e apoio próximos na realização deste trabalho. Em especial quero agradecer à minha família, em particular aos meus pais e avós, por me apoiarem nesta jornada académica. A Eles agradeço todo o carinho e motivação que sempre me deram.
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Contents
Resumo ...... i Palavras-Chave ...... i Abstract ...... iii Keywords ...... iii Acknowledgments ...... v Contents ...... vii List of Tables ...... ix List of Figures ...... x List of Symbols ...... xiii Abbreviations ...... xiv 1 Introduction ...... 1
1.1 Motivation ...... 1 1.2 Literature Review ...... 2
1.2.1 Multibody Formulation ...... 2 1.2.2 Biomechanics ...... 3 1.2.3 Application Case ...... 5
1.3 Thesis Organization ...... 6 1.4 Novel Aspects of the Work ...... 7
2 Multibody Dynamics Overview ...... 9
2.1 Cartesian Coordinates ...... 9 2.2 Kinematic Analysis ...... 10 2.3 Constraint Equations ...... 13
2.3.1 Joint Constraints ...... 13 2.3.2 Driving Constraints ...... 17
2.4 Equations of Motion ...... 21 2.5 Inverse Dynamic Analysis ...... 23 2.6 Joint Torques...... 24
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3 Biomechanical Model of the Human Skeletal System ...... 27
3.1 The Anthropometric Model ...... 27 3.2 Body Dimensions and Properties ...... 28 3.3 Model Topology ...... 29 3.4 Scaling Anthropometric Data ...... 31
4 Inverse Dynamic Analysis of Determinate Biomechanical Systems ...... 33
4.1 Data Acquisition ...... 33
4.1.1 Anthropometric Data ...... 33 4.1.2 Kinematic Data ...... 34 4.1.3 Kinetic Data ...... 36
4.2 Data Filtering ...... 38 4.3 Consistent Kinematic Data ...... 39 4.4 Application Case to a Normal Gait Cycle ...... 40 4.5 Validation of Results Using the Classical Newton-Euler Method ...... 43
5 Inverse Dynamic Analysis of Indeterminate Biomechanical Systems ...... 47
5.1 Muscle Actuators in Multibody Systems ...... 48 5.2 Dynamics of Muscle Tissue ...... 51 5.3 Lower Extremity Muscle Apparatus ...... 54 5.4 Optimization ...... 61 5.5 Application Case to a Normal Gait Cycle ...... 63 5.6 Discussion ...... 76
6 Conclusion and Future Development ...... 77
6.1 Conclusions ...... 77 6.2 Future Developments ...... 78
References ...... 81 Appendix A – Marker Set Protocol ...... 89 Appendix B – Visualization of the Biomechanical Model Using SAGA ...... 91
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List of Tables
Table 3.1: Anthropometric data of the sixteen anatomical segments. The lengths, Li and L̲i ,
and the distance to the centres of mass, di and d̲i , are schematically represented in Figure 3.2 (adapted from Rodrigo et al., (2008))...... 29 Table 3.2: Description of the joints used to interconnect the rigid bodies (adapted from Silva, (2003))...... 30 Table 5.1: Lower extremity muscle apparatus adjusted from Yamaguchi (2001). Described muscle information: action, maximum isometric force, pennation angle, resting and tendon length and attachment points. The illustration of each muscle was extracted from the site ‘Musculoskeletal Atlas’ and it is reproduced in this work with the permission of the authors...... 55
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List of Figures
Figure 1.1: Flowchart of the MATLAB program...... 8 Figure 2.1: Rigid body in Cartesian coordinates...... 9 Figure 2.2: Spherical joint between bodies i and j...... 14 Figure 2.3: Universal joint between bodies i and j. Representation of the kinematic constraint applied to the elbow of the human biomechanical model...... 15 Figure 2.4: Revolution joint between bodies i and j...... 16 Figure 2.5: Prescribed motion constraint applied to body i...... 17 Figure 2.6: Rotational driver constraint applied between body i and j...... 18 Figure 2.7: Spherical driver constraint applied between body i and j...... 19 Figure 3.1: Anthropometric model description. Illustration and table adapted from Silva (2003)...... 27 Figure 3.2: Representation of the anthropometric model. The annotations denote the body- fixed frames, the major dimensions and location of the centre of mass of each anatomical segment. (a) Perspective view in the standing position; (b), (c) Sagittal view of the head and neck; (d) Frontal view of the hand; (e) Frontal view of the upper and lower torso; (f) Sagittal view of the foot...... 28 Figure 3.3: Schematic representation of the sixteen rigid bodies and fifteen kinematic joints. (a) Description of the biomechanical model topology (adapted from Silva, (2003)); (b) Exploded view of the biomechanical model, including the identification of the forty-one DOF...... 30 Figure 4.1: Top view of the LBL using the QUALISYS program. Representation of the set of fourteen infrared cameras used to acquire the trajectories of the forty-four reflective markers placed on the subject’s skin and the three force plates used to acquire the ground reaction forces...... 34 Figure 4.2: Processing of the markers trajectory. (a) Male subject with reflective markers during a gait analysis at LBL; (b) Reproduction of the markers on the QUALISYS program; (c) Representation of the subject’s motion on MATLAB...... 35 Figure 4.3: Transmission of the ground reaction forces to the biomechanical system...... 36 Figure 4.4: Ground reaction forces. Representation of the three components of the ground reaction forces measured in the right and left foot during a complete gait cycle...... 37
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Figure 4.5: 2nd order Butterworth low pass filter. (a) unfiltered signal; (b) low pass filter representation; (c) filtered signal. (adapted from Winter (2009))...... 38 Figure 4.6: One gait cycle illustrated by the skeleton model of SAGA a) Sagittal view. right heel contact at 0%, right toe-off at 60% and again right heel contact at 100%; b) frontal view...... 40 Figure 4.7: Reaction forces in the right ankle, knee and hip joints obtained during one gait cycle analysis. a) X component. b) Y component. c) Z component...... 41 Figure 4.8: Joint torques developed in the sagittal plane at the right leg. The results obtained by Silva (2003) and Winter (2009) are also presented for comparison. a) hip b) knee and c) ankle...... 42 Figure 4.9: Free-body diagrams of the rigid bodies describing the a) foot, b) leg and c) thigh adapted. The unknowns are the forces and moments located in the proximal joint...... 44 Figure 4.10: Moment of force developed by in the sagittal plane at the right hip. The results were obtained using the multibody and analytical methods...... 45 Figure 4.11: Moment of force developed in the sagittal plane at the right leg: a) knee and b) ankle. The results were obtained using the multibody and analytical methods...... 46 Figure 5.1: Muscle actuators with and without via points. The semimembranosus, defined as a two-point muscle actuator and the tensor fasciae latae, defined as a four- points muscle actuator due to the muscle path complexity (adapted from Silva and Ambrósio, (2003))...... 48 Figure 5.2: Muscle actuator defined between its origin O and insertion I...... 49 Figure 5.3: Schematic representation of the dynamics of muscle tissue (based on Zajac, (1989))...... 51 Figure 5.4: Hill type muscle model used to simulate muscle contraction dynamics. The damping element is included in the contractile element (CE) and the series elastic element (SEE) is neglected...... 51 Figure 5.5: Muscle force properties considering a fully activated muscle...... 53 Figure 5.6: Lower extremity representation of the biomechanical model with the location of the reference frames of the local coordinates presented in Table 5.1. a) Isometric view of the lower extremity. b) Frontal view of the lower torso. c) Sagittal view of the lower torso...... 54 Figure 5.7: Biomechanical model with the lower extremity muscle apparatus applied in the right leg. Visualization generated in MATLAB...... 61
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Figure 5.8: Muscle activation patterns using two cost-functions: sum of the individual muscle forces (SINMF) (Collins, 1995) and sum of the cube of the individual muscle stresses (CIAMS) (Crowninshield and Brand, 1981). Feasible solutions were not found for the range of frames between 42% and 57% of the gait cycle...... 63 Figure 5.9: Activation patterns for the muscles of the locomotor apparatus. Comparison of two cost-functions: sum of the individual muscle forces (SINMF) (Collins, 1995) and sum of the cube of the individual muscle stresses (CIAMS) (Crowninshield and Brand, 1981)...... 69 Figure 5.10: Muscle forces for the muscles of the locomotor apparatus. Comparison of two cost-functions: sum of the individual muscle forces (SINMF) (Collins, 1995) and sum of the cube of the individual muscle stresses (CIAMS) (Crowninshield and Brand, 1981)...... 75
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List of Symbols
Convention a, A, α Scalar a Vector A Matrix
Over script
ȧ First time derivative ä Second time derivative ã 3x3 skew-symmetric matrix a̅ 4x4 skew-symmetric matrix
Superscript
aT , AT Matrix or vector transpose a′ Vector expressed in the body-fixed reference frame
Latin Symbols
A Generic transformation matrix a Muscle activation d Distance vector p Vector of Euler parameters f Vector of generic forces g Vector of generalized forces G 3x4 global transformation matrix I 3x3 Identity matrix J′ Inertia tensor L 3x4 local transformation matrix m Rigid body mass M Global mass matrix n′ Vector of local moments q, q , q Vector of coordinates, velocities and accelerations r Cartesian coordinates of generic point t Time x, y, z Global coordinates
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Greek Symbols
Φ Vector of kinematic constraints
Φ Jacobian matrix of kinematic constraints q λ Vector of Lagrange multipliers ν Right-hand-side vector of velocity equation
γ Right-hand-side vector of acceleration equation ξ, η, ζ Local (body-fixed) coordinates
Abbreviations
CE Contractile element CM Centre of mass EMG Electromyography DOF Degrees-of-Freedom ISB International Society of Biomechanics LBL Lisbon Biomechanics Laboratory PE Passive element SAGA System Animation for Graphical Analysis SEE Series elastic element
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1 Introduction
1.1 Motivation
The human locomotion apparatus consists of an articulated musculoskeletal system that is controlled by the central nervous system to perform all sorts of movements. From all the human movements, gait is an essential research field due to its importance in the daily life and requires a multidisciplinary approach to study the highly complex mechanism that is the human body (Cappozzo, 1984; Perry et al., 2010; Vaughan et al., 1999). The studies performed so far have led to a better understanding of the human biomechanics and encourage researchers to develop and improve mathematical models capable of simulating human activities with the objective of helping those who suffer from locomotion disorders (Fregly, 2009) or increasing the results of a high performance athlete (Rasmussen et al., 2012), for instance. The present work applies a multibody methodology to develop a three-dimensional biomechanical model of the whole body and to study the kinematics and dynamics of the lower limbs during one gait cycle. Inside the area of multibody biomechanics, researchers seek the continuous improvement of computational models (Arnold et al, 2010) to perform non-pathological gait as well as pathological gait analysis (Gonçalves, 2010), which can be caused by neurological disorders, physical deficiency or injury. The latter type of gait analysis also includes the study of the changes in the mechanical response when the subject is using an orthosis (Ferreira et al., 2013), an artificial external device that helps support the lower limbs, or a prosthesis (Kia et al., 2014), an artificial body part that substitutes an inexistent one, during a gait analysis. These methods are also developed to help surgeons assess a priori the consequences of a surgical intervention in patients suffering from diseases associated to pathological gait (Delp et al. 1990). The objective is to help the medical professionals make a better diagnosis of the patient’s condition, predict the aftermath of changing the biological system of the patient, and decide whether a surgical intervention or rehabilitation therapy should be performed (Ambrósio and Kecskeméthy, 2007). The implementation of biomechanical models to predict muscle forces based on a recorded movement using optimization procedures is preferred over in vivo measurements techniques, since the latter are invasive procedures and can only be applied to some muscles (Quental, 2013). Electromyography (EMG), specifically superficial EMG, provides muscle activation patterns of some muscles by placing electrodes in the subject’s skin (Tsikaros et al., 1997; Winter, 2009). However, there is some uncertainty among researchers in the results obtained using EMG (Bogey et al., 2005; Heintz and Gutierrez-Farewik, 2007) and these are mostly used to qualitatively validate the optimization methods applied to solve the muscle force sharing redundant problem.
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The objective of the present work is to implement a biomechanical model to study the human motion, namely, the human gait and to use a numerical robust methodology for the solution of its internal forces during prescribed motions. The computational model, developed in MATLAB, is capable of conducting non-pathological gait analyses, which are the focus of the application case of this work, by assessing the joint torques developed in the articulations as well as estimating the muscle forces in the lower limb.
1.2 Literature Review
Researchers are focused on the development of more accurate mathematical models to simulate the human body motion and predict the mechanical behaviour during specific human activities. Initially, gait analyses considered motion only in the sagittal plane, i.e., using two-dimensional partial representations of the human body. Over the last decades, with the increase in computational power, three-dimensional models have been developed with a higher level of detail (Rajagopal et al., 2016).
1.2.1 Multibody Formulation
The mathematical formulation employed depends on the area of research, the objective of the analysis and the results expected (Silva, 2003). If the objective is to analyse in detail the localized structural deformation or the mechanical behaviour of structures, finite element methods are usually used (Quental, 2013). Otherwise, if large displacements and complex interactions with the surrounding environment are to be analysed, multibody models are commonly applied (Silva, 2003). In a multibody model, the mechanical characteristics of the bodies, mass and inertia, and of the joints, type and location, are more important than the bodies geometries, which are only required when contact detection is required. Hybrid models (Monteiro et al., 2011), which consist in the assemblage of multibody and finite element models, are also an option for analysis addressing very specific objectives. By implementing a co-simulation, of multibody and finite element, it is possible to join the advantages of both mathematical formulations with the cost of a higher computational complexity. The multibody simulation of the human biomechanics during a gait analysis provides valuable information concerning the external forces, generated by contact with the surrounding environment, and the internal forces acting between the rigid bodies that compose the biomechanical system. These results can be obtained by formulating and solving either a forward or an inverse dynamic analysis. The former requires the initial kinematic conditions of the mechanical system, i.e., the initial position and velocity of each body, and the externally applied forces to provide the dynamic response of a constrained multibody system. The latter, which is the one applied in this work, is used when the motion of the mechanical system and the externally applied forces and moments are known, and the objective is to determine the unknown forces and moments that cause that particular motion (Nikravesh, 1988).
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Depending on the multibody system to be studied there are two sets of coordinates that can be used: independent and dependent coordinates. Independent coordinates represent the kinematic chain of a mechanism with a minimum number of coordinates, generally equal to its number of degrees-of- freedom (DOF). However, the equations of motion have a high nonlinearity being more difficult to implement computationally. Current commercial or public available codes, based on independent coordinates are OPENSIM (Delp and Loan, 1995) or MADYMO (T.N.O, 1997). The dependent coordinates can be subdivided into three types: joint coordinates, fully Cartesian or natural coordinates and Cartesian coordinates. The joint coordinates define the orientation of each moving body with respect to another body that can be moving or not and, therefore, these coordinates are also called relative coordinates,6 being minimum for open loop systems (Nikravesh, 1988). Although the application of joint coordinates results in a lesser number of coordinates than the application of the other two sets of dependent coordinates, its formulation has a higher order of nonlinearity that makes their computational implementation complex (Jálon and Bayo, 1994). The fully Cartesian coordinates define the position and orientation of bodies using points and unit vectors, and consequently no angular parameters are required, while with the Cartesian coordinates, these are defined by three Cartesian coordinates and three or four angular parameters, depending on the angular formulation applied. The former is more advantageous to study the kinematics of biomechanical systems (Pereira & Ambrósio, 1994; Jálon and Bayo, 1994; Jálon, 2007), since the bodies are defined using the joint centres, which are measured directly in a biomechanical laboratory. However, the application of Cartesian coordinates is very intuitive in general purpose programs, being the equations of motion, including the global mass matrix and external applied forces vector, assembled in a straightforward way. Hence, in this work, Cartesian coordinates were selected to build the biomechanical model of the human body.
1.2.2 Biomechanics
The human multibody system is a representation of the complex human musculoskeletal system. The anatomical segments correspond to rigid bodies, the articulations are represented by geometrically ideal or contact mechanical joints and the muscles are described by mechanical actuators. Within the multibody models, there are two-dimensional (Blajer et al., 2007) and three-dimensional models (Silva, 2003). The implementation of two-dimensional models is viable if the movement under analysis is mainly performed in a plane, as for instance, in a gait or cycling analysis. However, the projection into the sagittal plane causes the loss of some movement characteristics and precludes the estimation of the joint torques and muscle forces in the remaining planes (Brändle, 2016; Umberger & Martin, 2001). The human biomechanical model can consist in a detailed part of the body (Malaquias et al., 2015; Ribeiro et al., 2012), a representation of the upper or lower limbs (Arnold et al., 2010; Quental, 2013)
3 or a full body description (Silva, 2003). If the objective of the analysis only concerns the mechanical response of a specific limb, a partial description of the human body can be more efficient and less computational demanding than a whole body description. Usually for gait analyses, the biomechanical models used are composed of the lower limbs and an aggregation of the upper anatomical segments, head, arms and trunk, commonly named HAT, in a single rigid body (Horsman, 2007; Winter, 2009; Yamaguchi, 2001). Nevertheless, one purpose of this work is to implement a model that can be used to eventually take into consideration the influence of the upper limbs motion in the solution of the inverse dynamic analysis. With the continuous development of the biomechanical models, whole body models with a detailed representation of particular anatomical segments and articulations, such as the foot and the femur-patella joint, respectively, are expected to be presented in the future (Ambrósio, 2007). In this work, a three-dimensional whole body biomechanical model composed of sixteen rigid bodies and fifteen joints is defined. The anthropometric model applied is based on the combination of two models: a computer simulation code SOM-LA (Laananen, 1991; Laananen et al., 1983) and a general-purpose model (Celigüeta, 1996). The second anthropometric model provides data regarding the feet, hands and head of the human body, while the rest of the human body is described by the first model. The anthropometric data was obtained from the work of Chandler et al. (1981) considering the mass distribution and body size of the 50th percentile male. The anthropometric model and data were both compiled by Silva (2003) and are reproduced in this work. The musculoskeletal system (Nigg and Herzog, 1999; Zatsiorky and Prilutsky, 2012) applied to the biomechanical model to study the human locomotion apparatus only covers the lower muscle extremity apparatus. The muscles are introduced into the biomechanical model as actuators with geometric and physiological properties. There are few muscle databases reported in the literature describing the muscle properties of the lower extremity (Arnold et al., 2010; Carbone et al., 2015; Horsman, 2007; Rajagopal et al., 2016; Yamaguchi, 2001) since the measures are based on cadaver studies. These databases were built to be incorporated in biomechanical software applications, such as ANYBODY (Damsgaard et al., 2006), OPENSIM (Delp et al., 2007) and Software for Interactive Musculoskeletal Modelling (SIMM) (Delp et al., 1990; Delp and Loan, 1995). The muscle database presented in the work of Yamaguchi (2001) was selected to allow a direct comparison of results with those obtained by Silva (2003). This muscle database has all the muscle properties required to define the muscle actuators and build an independent program in MATLAB. The lower muscle extremity apparatus considered is composed of forty-three muscles per leg. The mechanical behaviour of the muscle actuators is described by the Hill-type model that is the most used in muscle-driven simulations (Yamaguchi, 2001). This model encloses the tendon and the muscle fibre contraction dynamics, which provides the muscle force produced knowing the muscle activation. The activation dynamics that represents the lag between the central nervous system decisions
4 and the activation of muscles is neglected. The tendon can be modelled as an elastic or a rigid element (Millard et al., 2013; Oliveira et al., 2015) depending on the human task under analysis and the length of the tendon. For slow motions and short tendon lengths, the rigid tendon model is usually implemented (Quental et al., 2016). Additionally, comparing short tendon simulations, the rigid-tendon model simulation is faster and provides the same muscle forces as those calculated by the elastic-tendon model (Millard et al., 2013). Although some lower extremity muscles have large tendon lengths, in this work, only the rigid-tendon model is considered to simplify the muscle modelling. This assumption implies that part of the muscle dynamics, or muscle contraction dynamics, associated with the tendon deformation is not considered here. Therefore, faster motions than gait, such as jumping, should not be considered.
1.2.3 Application Case
To perform an inverse dynamic analysis, the motion of the multibody system to be studied must be known in advance as well as the externally applied forces in order to calculate the remaining forces and moments (Nikravesh, 1988). Hence, for a gait analysis, kinematic and kinetic data of the subject under analysis must be gathered. The kinematic data is acquired using reflective markers placed on the subject’s skin, which are traced by infrared cameras to obtain the anatomical segments position and orientation throughout the gait cycle. The location of the anatomical markers follows the recommendations of the International Society of Biomechanics (ISB) (Wu et al., 2002, 2005). The kinetic data is collected by three force plates placed on the ground which record the ground reaction forces and the centre of pressure on the feet. All data acquisition equipment is available at the Lisbon Biomechanics Laboratory (LBL), which is used to support the work now presented. In this work, two inverse dynamic analyses are performed to achieve two types of results: calculation of the joint torques developed at the articulations, which are responsible for guiding the DOF of the system, and the muscle forces developed in the lower extremity apparatus. The joint torques are obtained by solving the equations of motion directly, since the number of unknowns is equal to the number of equations of motion. On the other hand, the determination of the muscle forces is not straightforward, since the biomechanical model has more muscles than DOF. Therefore, two different problems are addressed, i.e., the inverse dynamic analysis of a determinate and an indeterminate biomechanical system, respectively. The indeterminate biomechanical problem is solved through an inverse dynamic approach, but it requires the use of a different procedure to deal with the excess of muscles in comparison to DOF. There are several methods that can be applied to obtain a feasible solution. Two simple methods consist in the shortening and enlargement of the number of muscle constraint equations and number of equations of motion, to respectively, obtain an equal number of constraints and equations of motion. However, in the
5 first case, the detail of the muscle apparatus is lost and in the second the evaluation of additional equations becomes more difficult (Tsikaros et al. 1997). In gait analyses, optimization methods are commonly used to obtain the muscle activations. There are several types of optimization methods (Ackermann, 2007), but the most common are: static optimization and dynamic optimization. The static optimization provides, for each instant of time, the set of muscle activations that minimizes a physiological criterion while respecting the equations of motion of the biomechanical model and the physiological boundaries of the muscle activations (Quental et al., 2016). This method is based on an inverse dynamic approach that finds the muscle activations responsible for the observed motion. On the other hand, the dynamic optimization uses a forward dynamic approach with time-dependent variables to find the muscle forces that generate the motion under analysis (Anderson and Pandy, 2001a). Although the dynamic optimization is more powerful and accurate than the static optimization, it is also much more expensive computationally, which often compromises its application (Anderson and Pandy, 2001a). Recently, a new optimization process was presented, named window moving inverse dynamics optimization, which overcomes some limitations of the static optimization while increasing its computational cost just by an affordable increment. The interested reader is referred to the work of Quental et al. (2016) for further information. Considering the muscle model applied here, the static optimization is the most efficient of the three and its results are equivalent for gait analyses to those obtained using the dynamic optimization (Anderson and Pandy, 2001b). Therefore, in this work, the static optimization method is applied to calculate the muscle forces of the biomechanical model.
1.3 Thesis Organization
The outline of the present thesis is described here. Chapter 1 provides a context of the analysis performed in this work, a review of the current developments concerning the human locomotion apparatus and an overview of the work developed. The multibody methodology required to perform an inverse dynamic analysis is explained in detail in Chapter 2. The kinematic and driver constraint equations, as well as the equations of motion, are described in this chapter. In Chapter 3, the human biomechanical model developed is presented. Details regarding the dimensional and physical properties of each anatomical segment, the system topology and the procedure applied to scale the anthropometric data are provided in this chapter. The following two chapters, Chapters 4 and 5, present the results for the two types of inverse dynamic analysis performed, i.e., determinate and indeterminate inverse dynamics of biomechanical systems, respectively. Chapter 4 addresses, first, the procedures followed to acquire the anthropometric, kinematic and kinetic data of the human gait. Then, the implementation of a kinematic analysis is
6 presented to ensure the calculation of consistent positions, velocities and accelerations. The chapter ends with the presentation and discussion of the inverse dynamic analysis results. The indeterminate biomechanical system, presented in Chapter 5, uses the same input data of Chapter 4, but the aim of the inverse dynamic analysis performed is to obtain the muscle forces developed in the lower limbs. After describing the multibody representation of the muscle actuators and the respective dynamic model used, the lower extremity muscles implemented in this analysis, as well as the method developed to scale them, are presented. This is followed by the description of the optimization procedure applied to obtain the muscle activations and consequently the muscle forces. The chapter closes with the solution and discussion of two optimization results obtained using two criteria: sum of the square of the individual muscle forces and sum of the cube of the individual average muscle stresses. The conclusions of this work and future work developments are presented in Chapter 6. Besides the described chapters, the present thesis also includes appendices in which the marker set protocol developed to perform the motion acquisition of human gait and a graphical visualization tool of the biomechanical model motion is described.
1.4 Novel Aspects of the Work
The present thesis uses as background knowledge the work of Silva (2003), in which a whole body biomechanical model is implemented to study the human gait, but instead of using a fully Cartesian formulation, a Cartesian formulation was considered. The biomechanical model, adapted to a Cartesian formulation, and the methods required to perform an inverse dynamic analysis were implemented in MATLAB. Both the determinate and indeterminate problems were considered in order to calculate the joint torques and the muscle forces in the lower limbs. To reach these results, an acquisition marker set protocol, based on the recommendations of the International Society of Biomechanics (ISB) was developed, to generate the kinematic data input. Figure 1.1 illustrates the data flow of the MATLAB program with the respective inputs and outputs identified. Besides the visualization feature developed in the MATLAB environment to check the human movement and the muscles position, three human models were also adapted in SOLIDWORKS to be used in the simulation tool ‘System Animation for Graphical Analysis’ (SAGA) (Milho and Ambrosio, 1995): male, female and skeleton. This tool is essential to make a visual assessment of the human motion. The numerical aspects to deal with critical issues on the numerical solution of the inverse dynamics problems using Cartesian coordinates are also addressed. A rotational driver for spherical joints is implemented.
7
Human male Pre-process data Kinematic Analysis Biomechanical model • Filter anatomical markers Φq ,0t Motion Capture & and ground reaction forces Force plates • Define local axes Φq qν • Determine joint angles Lower extremity • Scale anthropometric data Φq qγ muscles external forces
Inverse Dynamic Visualization Static Analysis tool Optimization T Mq Φλgq
SAGA input file Muscle Forces Joint000 Moments
Figure 1.1: Flowchart of the MATLAB program.
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2 Multibody Dynamics Overview
The multibody formulation used to study the motion of the human body is presented in this chapter. This formulation is used to model a three-dimensional multibody system and the corresponding methodology is presented here to perform a kinematic and inverse dynamic analysis. A multibody system consists in an aggregation of rigid bodies, interconnected by kinematic pairs or joints and acted upon by externally applied forces (Silva, 2003). From all the sets of coordinates, Cartesian coordinates were the one selected to support the model implementation due to the simplicity of the formulation. The notation adopted in this work closely follows that used by Nikravesh (1988).
2.1 Cartesian Coordinates
An unconstrained three-dimensional body has 6 degrees-of-freedom (DOF) and is globally defined by three translational and three rotational coordinates. The translational coordinates locate the origin of the local reference frame, also called body-fixed reference frame, in relation to the global reference frame and the rotational coordinates define the body orientation. Figure 2.1 illustrates a general rigid body configuration with Cartesian coordinates.
P
i P P si ri
Y i i (i) ri
X Z Figure 2.1: Rigid body in Cartesian coordinates.
Alternatively, to the use of three rotational coordinates, commonly represented by Euler angles, the body orientation is defined by four coordinates known as Euler parameters. Both types of rotational coordinates lead to the same rotation matrix A. However, the Euler parameters are free of some of the deficiencies of the Euler angles, the most concerning of which being singularity issues (Nikravesh, 1988). Accordingly, the application of Euler parameters is preferred over the Euler angles. The four Euler parameters represent a rotation about a specific axis and are denoted by p. A special property of the Euler parameters is the equation pTp = 1, which allows to obtain a fourth parameter if three parameters are already known. Using Cartesian coordinates, the definition of point P of body i in space, is given by:
9
PP riiii r A s (2.1)
P where ri are the global coordinates of point P located in body i, ri are the global coordinates of the
P body-fixed reference frame, Ai is the rotation matrix of body i, defined by the Euler parameters, and s′i are the local coordinates of point P. The configuration of a multibody system in any instant of time is defined by the vector of coordinates q that contains the coordinates of each body. If a multibody system comprises n bodies, the resultant vector q has (7xn) entries:
T qqqqqrpTTTTT,,,with,,,,,,, x y z eeee T (2.2) 120123 niii i
where ri denotes the translational coordinates and pi the Euler parameters of a body i. The kinematic pairs constrain the relative motion between the bodies, decreasing the number of DOF of the multibody system. Each individual joint introduces a relationship between the coordinates of the bodies, resulting in a vector of kinematic constraint equations. Consequently, the number of DOF of the multibody system decreases by the same number of independent constraint equations, imposed by the joint. All the constraint equations must be fulfilled for every instant of time, i.e.:
T Φ qΦ,,,,,,ttt qΦqΦ ,,,11 qΦq0nsnr (2.3) where Φi corresponds to the kinematic constraint equations of joint i, ns is the total number of scleronomic constraints, nr is the total number of rheonomic constraints and 0 is a null vector. Scleronomic constraints are time independent, i.e., the time variable does not appear explicitly in the algebraic equations, whereas rheonomic constraints are time dependent. The first type is usually related to kinematic pairs, while the latter is associated to driver actuators (Silva, 2003).
2.2 Kinematic Analysis
By definition, the kinematic analysis consists in the description of the multibody system motion without considering the body masses or any external forces that could be applied to the bodies (Nikravesh, 1988). Since external forces are not considered, the motion of the mechanical system must be specified. The DOF of the system are controlled by driving constraints that prescribe the position, velocity and acceleration of some elements, while the position, velocity and acceleration of the remaining elements are obtained using the kinematic constraint equations that describe the topology of the system (Silva, 2003). The kinematic consistent positions are obtained by solving Equation (2.3) with respect to the vector of coordinates q, which means that the positions satisfy the kinematic constraint equations. The constraint equations are non-linear, and thus the system of non-linear equations must be solved using an
10 appropriate method. Although, a MATLAB function based on optimization methods, called fsolve, is used in this work to find the solution of nonlinear systems as well as linear systems of equations, the Newton-Raphson method is presented here as an example to address a more general basis for computer implementation. This method is based on the linearization of Equation (2.3), which consists in replacing the system of equations with the first two terms of its expansion in a Taylor Series, assessed at an initial approximation vector qi. Accordingly, Equation (2.3) becomes:
Φ q,,tt Φ qi Φq q i q q i 0 (2.4) where Φq(qi) is the Jacobian matrix of the constraints, evaluated at the approximate solution qi. The iterative process is stopped when the residual, defined as ∆qi = qi+1 − qi, is less than a predefined tolerance. The Jacobian matrix of the constraints is defined by the partial derivatives of each kinematic constraint with respect to the vector of coordinates, resulting in the mathematical expression:
11 qq 1 nc Φqq (2.5) nhnh qq1 nc where nh and nc are the number of constraints and rigid bodies, respectively. The kinematic consistent velocities of the elements that constitute the mechanical system are calculated using the velocity constraint equations, determined by differentiating Equation (2.3) with respect to time:
Φ(, qΦ )(,tt q ) Φ q,, qq0t t q (2.6) where the term ∂Φ(q,t)/ ∂t is the vector containing the partial derivatives of the constraints with respect to time, the term ∂Φ(q,t)/ ∂q is the Jacobian matrix, already defined in Equation (2.5), and q is the vector of velocities. Defining vector ν(t) as the right hand side vector of the velocity equations, the velocities of the system are obtained by the solution of the following expression:
Φq(,)t Φq q ν, with ν (t ) t (2.7)
Note that only rheonomic constraints, associated with driver equations, contribute with non-zero entries to this vector.
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The kinematic consistent accelerations are found following the same method used in the velocity analysis. The acceleration constraint equations are obtained by a double differentiation of Equation (2.3) or by the differentiation of the Equation (2.6), which leads to:
ΦqqqΦqΦqqν0 ,,,() tt qq q (2.8)
The vector γ(t) is defined as the right hand side vector of the acceleration equations and the accelerations of the system are obtained by solving the expression:
Φqγγqqq,with(,,)( qνΦqq ) tt q (2.9)
In the context of the data processing of the acquired kinematic data for the inverse dynamic analysis of the biomechanical model there are two different methods to calculate the velocity and acceleration vectors, when the consistent positions vector is known. The first method consists in solving the Equations (2.7) and (2.9) with respect to the velocity and acceleration vectors, respectively. The second method consists in interpolating the trajectories of each global coordinate using cubic splines and then differentiating them with respect to time to obtain their velocity and acceleration. Although the second approach does not guarantee the exact fulfilment of the velocity and acceleration equations, in the context of the inverse dynamic analysis of biomechanical models for gait analysis, both methods are used in this work and give similar results. Nonetheless, it must be noted that their concept is different. The first uses the velocity and acceleration equations, meaning that the kinematic consistency is guaranteed, whereas the second considers that the differentiation of the kinematic consistent positions does not lead to kinematic consistent velocities and accelerations, although they are close enough to the consistent ones. A final remark must be made regarding the possible existence of redundant constraints in Equation (2.3). If the multibody system under analysis has redundant constraints, i.e., more equations than DOF, the resultant Jacobian matrix has linearly dependent lines. Different methods can be applied to overcome this issue. The first method is to use the least-squares formulation, which consists in the pre-multiplication of both sides of the equations by the transpose of the Jacobian matrix, as proposed by Jalón and Bayo (1994). The second option, and the one used in this work, is the MATLAB function backslash. This function provides a solution in the least square sense, similar to the first method, by generating a pseudo-inverse matrix. However, when using pseudo-inverse methods, the results must be handled with caution because a solution is always found, even if it has no physical meaning.
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2.3 Constraint Equations
The constraint equations are presented in a general form and are not dependent on the type of coordinates used. The purpose of this section is to introduce the relevant types of kinematic constraints needed to build the human biomechanical system used in this work. For further description of other types of kinematic constraints, the interested reader is referred to Nikravesh (1988). There are essentially two types of kinematic constraints when using Cartesian coordinates: joint constraints, which constrain the relative motion between two bodies and driver or driving constraints, which prescribe the motion between two bodies or guide the motion of a single body. The constraint equations and the contribution to the Jacobian matrix and right-hand side vectors of the velocity and acceleration equations are presented here for both types of constraints. Before addressing the different types of kinematic constraints, note that the Euler parameters are ideal to represent the angular orientation of a body, but they generate too many equations when using their time derivatives. Accordingly, to reduce the number of equations, the Jacobian matrix and right hand side vectors of the velocity and acceleration equations are modified to be described with respect to the three local angular velocities ω′ and accelerations ω , instead of the four velocities ṗ and accelerations p . It must also be noted that by proceeding in this manner the mathematical constraint
T pi pi = 1 does not need to be used. This procedure, explained in detail in the work of Nikravesh (1988), preserves the advantages of the Euler parameters properties (Nikravesh, 1988). As a result of this modification, the vector of velocities q and accelerations q have only six entries for each rigid body.
2.3.1 Joint Constraints
The joint constraints described here are applied to establish relative motion constrains between two rigid bodies of the human biomechanical model. The model is composed of spherical, universal and revolute joints.
2.3.1.1 Spherical Joint
A spherical joint between two rigid bodies i and j establishes three algebraic equations that constrain the position of a shared point P, located at the centre of the spherical joint. If one of the bodies is fixed, only three relative rotational DOF exist between the bodies. The general configuration of this joint is illustrated in Figure 2.2 and the constraint equations are expressed as:
(s,3) PP Φ ri Α i s´´ i r j Α j s j 0 (2.10) where the superscripts in Φ indicates the type of joint and the number of constraint equations, r is the global position vector, A is the rotation matrix and s′P is the local vector associated to either body i or j.
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To reduce the number of equations, the Jacobian matrix and right hand side vectors of the velocity and acceleration equations are modified so that their entries are described by three angular velocities. The differentiation of Equation (2.10) with respect to time leads to (Nikravesh, 1988):
(s,3) (s,3) (s,3) (s,3) (s,3) ΦΦΦΦ Φ ri p i r j p j ri p i r j p j
(s,3) (s,3) (s,3) (s,3) (2.11) (s,3) ΦΦΦΦ11 TT Φ ri L i ω i r j L j ω j ri22 p i r j p j where ∂Φ(s,3)/ ∂r and ∂Φ(s,3)/ ∂p are the vectors containing the partial derivatives of the constraints with respect the position r and the Euler parameters p, respectively, and L is a local transformation matrix that is defined in Nikravesh (1988). The contribution of the spherical joint to the Jacobian matrix and the right hand side vector of the velocity equations are expressed as (Nikravesh, 1988):
r11ω ri ω i r j ω j r n ω n (s,3) PP Φq 00 IsA i i IsA j j 00 (2.12) ν0(s,3) where I is the 3x3 identity matrix and s͂P is the skew-symmetric matrix of the vector sP of either body i or j, expressed as (Nikravesh, 1988):
0ss zy P (2.13) s s0szx ss0yx
In Equation (2.12), the reference for the columns associated to entries of the vector of velocities are presented above the Jacobian matrix. The Jacobian matrix format is kept throughout this section to demonstrate the position of the non-null contributions.
j i i P s P si j j i P j
(i) (j)
rj ri Y
Z X
Figure 2.2: Spherical joint between bodies i and j.
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The second time differentiation of Equation (2.11) leads to the acceleration equations out of which the right hand side vector is written as (Nikravesh, 1988):
(s,3) PPPP γω sωssωs iijj with (2.14) where ῶ is the skew-symmetric matrix of the global angular velocity and ṡP is the time derivative of the vector sP of either body i or j.
2.3.1.2 Universal Joint
A universal joint between bodies i and j is shown in Figure 2.3 and illustrates the kinematic constraint applied to the elbow of the human biomechanical system. The cross in the middle of the bodies can be associated to body i and it defines two axes of rotation that intersect at point P.
i
(i) i
i
P Q si si ri
Y Qi si P X s Z j
Qj P s j rj Q s j
j (j) j j Figure 2.3: Universal joint between bodies i and j. Representation of the kinematic constraint applied to the elbow of the human biomechanical model.
The two axes represented by vectors si and sj , which are arbitrarily placed along each axis, must always remain perpendicular. This joint can be seen as a combination of two kinematic constraints: a spherical joint located at point P and two perpendicular vectors. The constraint equations of the universal joint are:
(s ,3) (u ,4) Φ Φ0T (2.15) ssij
15
Between the two bodies there are only two DOF associated with each of the rotation axis defined. The contribution of the universal joint constraints to the Jacobian matrix, as well as to the right hand side vectors of the velocity and acceleration equations, can be written as (Nikravesh, 1988):
(s,3) (u ,4) Φq Φq TT 000s s...... A0s s A00jiiijj ν0(u ,4) (2.16) (s ,3) (u ,4) γ γ TTT 2sijiijjji ss ω ss ω s
2.3.1.3 Revolute Joint
A revolute joint between the bodies i and j is shown in Figure 2.4. Similarly to the universal joint, this joint also has a spherical constraint located at point P. However, it is constrained by two more equations, which means it only has a rotational DOF about the axis defined.
i i
(i)
i
ri Q P si si Y Q Qi j s j P X s Z i
P Q s s j rj j si 2 j (j) si1 j j
Figure 2.4: Revolution joint between bodies i and j.
The joint axis of rotation is defined by vectors si and sj , defined in bodies i and j, respectively. The rotational constraint equations must ensure that these two vectors always remain parallel. However, this leads to three constraint equations, one of which is linearly dependent of the remaining. To overcome this indeterminacy, instead of using a single cross-product, the constraint is unfolded into two dot-products (Quental, 2013). Figure 2.4 illustrates, in the left low corner, the generation of two additional vectors si1 and si2 , which are both perpendicular to si and to each other. The constraint equations of the revolute joint are expressed as:
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Φ(,3)s (,5)rT Φss0 ij1 (2.17) T ssij2
The contribution of the revolute joint constraints to the Jacobian matrix and to the right hand side vector of the velocity and acceleration equations is given by (Nikravesh, 1988):
(s,3) Φq (r ,5) TT Φq 00... 0ssAj i11 i ... 0ssA i j j ... 00 TT 00... 0ssAj i22 i ... 0ssA i j j ... 00 ν0(r ,5) (2.18) γ(s ,3) (r ,5) T T T γ 2 si1 s j s i 1 ω i s j s j ω j s i 1 TTT 2si2 s j s i 2ω i s j s j ω j s i 2
2.3.2 Driving Constraints
The driving constraints here described are used to guide the DOF of the human biomechanical model, specifically, the DOF of the whole body and joints. In general analysis of biomechanical models, the guidance of the intersegmental angles requires the use of this type of constraints.
2.3.2.1 Prescribed Motion Constraint
A prescribed motion constraint is applied to control the position and orientation of a body, as illustrated in Figure 2.5.
i
i
Y ri * qii( tq )( ) t i (i) X Z Figure 2.5: Prescribed motion constraint applied to body i.
In this work, this constraint is applied as a driver constraint to control the motion of the base body and, consequently, guide the whole biomechanical system. The constraint equations are written as:
(pmc ,7) * Φ qii q()t 0 (2.19)
17
* where qi (t) is a time dependent position and orientation vector that body i must follow for every instant of time. Note that there are seven constraints to control the six DOF of a body, which means there is one redundant constraint. To deal with the redundancy, the least square formulation discussed in Section 2.2 was applied. The contribution of the prescribed motion constraint to the Jacobian matrix and to the right hand side vectors of the velocity and acceleration equations is:
(,7)pmc 00I000...... Φq 00000 ...... 1 T 2 Li (,7)*pmc νq i ()t (2.20) * ri ()t γ(,7)pmc * 1 T piiii()t ωωp 4
* * * where q̇i (t), ṙi (t) and ṗi (t) represent the time dependent vectors of the velocity, translational acceleration and rotational acceleration of Euler parameters associated with body i. These vectors are
* determined by interpolating the prescribed motion vector qi (t) using cubic splines and by differentiating these once and twice with respect to time.
2.3.2.2 Rotational Drivers
A rotational driver constraint is applied between two bodies to control the motion of one or more rotational DOF. In this work, these constraints are applied between bodies that are connected by a kinematic pair and the objective is to guide the remaining DOF. Figure 2.6 exemplifies the application of this constraint for a single rotational DOF between two bodies.
(i)
i Y i ri i u X ()t Z rj j v j
j
(j)
Figure 2.6: Rotational driver constraint applied between body i and j.
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The mechanical motion of the human knee can be described by a revolute joint and a rotational driver. The kinematic joint allows only the knee to flex and extend and the driver is responsible for controlling the angle variation θ(t). The vectors u and v are defined as unitary vectors and, in this case, u has the same direction of the axis ηi and v has the opposite direction of ηj. Before performing a kinematic consistent position analysis, the vectors u and v are defined for each instant of time and the variation of the angle θ(t) is determined. Since both vectors are unitary, their dot-product is equal to the cosine of the angle between them. Consequently, the constraint equation is written as:
Φuvu(,1)rdT v cosθ(ttt )0withθ( ),( ) (2.21)
This driver can be applied twice to control the two DOF of a universal joint placed between two bodies. The contribution of the driver constraints to the Jacobian matrix and the right hand side vectors of the velocity and acceleration equations is given by:
(,1)rdTT Φ000vq uA0u vA00...... ij ν(,1)rd sinθ(tt ) cos(θ( )) (2.22) (,1)2rdTTT γu vu ω2cos vv ω u ijθ(tttt ) sin(θ( ))sin(θ( ))θ( ) where θ( t) and θ( t) are, respectively, the time dependent vectors of the velocity and acceleration of the angle between the unitary vectors u and v. These vectors are computed as the first and second derivatives of a cubic spline that interpolates the vector θ(t). Note that the direction of vectors u and v must be chosen to ensure that the angle θ(t) varies between [00,1800], i.e., the range of motion of the DOF varies inside this interval, since the angle θ(t) is calculated using the inverse cosine function. The spherical driver is responsible for guiding the DOF of a spherical joint. Figure 2.7. illustrates a typical spherical joint with a rotational driver applied to guide three angles that correspond to the three rotational DOF. The idea here is to control, using a kinematic constraint, the three independent angles about their respective instantaneous rotation axis between two bodies connected by a spherical joint. Hence, the complete relative motion between these two bodies is fully controlled.
()t
i j d c j b i i u ()t a j (i) v (j)
()t Figure 2.7: Spherical driver constraint applied between body i and j.
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When used in the framework of inverse dynamic analysis of deterministic biomechanical models the joint reaction forces, or moments in this case, are the joint driving torques about the anatomical joint represented by a spherical joint. Similarly to the rotational driver constraint presented previously, the angles formed by the vectors α(t), β(t) and θ(t) are pre-calculated before determining the kinematic consistent positions. The constraint equations are established considering dot-products between the unitary vectors that are parallel to the local axes of the bodies local reference frames, ensuring in this form the fixed alignment between body fixed frames. The procedure used here has similarities with that used by Pombo and Ambrósio (2003) in the definition of the path following constraint, or by Viegas (2016) in the definition of the vehicle rail following constraint in roller-coaster dynamics. Considering that the vectors a, c and u are associated with body i while the vectors b, d and v are associated with body j. For the sake of conciseness let it be assumed that a, c and u are parallel to the unit vectors along the body fixed frame ui, ui and ui, respectively, and b, d and v are parallel to the unit vectors along the body fixed frame uj, uj and uj, respectively. Mathematically, these are expressed as:
uuT cosα(t ) ij Φuu0(,3)sdT cosβ(t ) (2.23) ij uuT cosθ(t ) ij
Note that the vectors that define the alignment about the spherical joint do not need to be parallel to the body fixed frames. It is only required that these vectors are fixed to each one of the bodies and, for those fixed to the same body, orthogonal to each other. Then, the constraint defined by these equations α(),()tt ab , β(),()tt cd and θ(),()tt uv , correspond exactly to Equation (2.23). The velocity constraint equations of the spherical driver constraint are the time derivatives of the position constraint equations, expressed by:
uTT uu u sinα(t ) α i jj i (sdTT ,3) u uu u0 sin β(t ) β (2.24) i jj i TT u uu u sinθ(t ) θ i jj i which by substituting u = Ȧu' and by using the identity Ȧ = Aῶ', and rearranging becomes:
uTTTT A A u u A A u sinα(t ) α ii j j j j j i i i (sd ,3) u T A T A u u T A T A u sin β(t ) β 0 (2.25) ii j j j j j i i i TTTT u Ai A j u j u A j A i u i sinθ(t ) θ i j j i
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(,3)(,3)(,3)sdsdsd which is finally written in the matrix form q q :
r i 0uA ATTTT u0uA A u sinα(t ) α jiij jiij i (,3)sdTTTT0uA A u0uA A u sin β(t ) β 0 (2.26) jiij jiij TTTT 0uA A u0uA A u rj sinθ(t ) θ jiij jiij j
( ,3)sd In Equation (2.26) the Jacobian matrix associated to the spherical driver constraint, q , and the right hand side of the velocity equation, (sd ,3) , are clearly identified by similarity. The acceleration constraint equation associated to the spherical driver constraint, is obtained by the time derivative of the velocity constraint Equation (2.26), i.e.,
r i 0u AT A u0uTT T A A u jiij j ii j i (sdsdsdT ,3)( ,3)( TT ,3) T q0u A A u0u A A u q jiij j ii j T TT T 0u A A u0u A A u rj jiij j ii j j (2.27) T TTT 2 u iii A AA jii AA2cos jji Au jjj α(tt ) α sin α( ) α i j uATTTT AA AA Au 2cos β(tt ) β sin β( ) β2 0 ij iii jii jji jjj uATTTT AA AA Au 2cos θ(t )θ sin θ(t ) θ2 ij iii jii jji jjj where α( t) , β( t) and θ( t) are the velocities and α( t) , β( t) and θ( t) are the accelerations of the angles α(t), β(t) and θ(t), respectively. Considering a cubic spline interpolation for each angle, the velocity and acceleration are obtained by differentiating the computed interpolation with respect to time. The right hand side contribution of the acceleration constraint equations associated to the spherical driver torque to the equations of motion, (sd ,3) , is easily identified in Equation (2.27) by similarity.
2.4 Equations of Motion
The equations of motion of a constrained multibody system can be obtained using one of several formulations: Lagrange’s equations; principle of virtual work; or the principle of virtual power (Silva, 2003). In the present work, they are obtained by using the principle of virtual power (Jalón and Bayo, 1994), which states that the sum of the virtual power produced by the inertial and external forces acting on the mechanical system must be zero in any instant in time. Mathematically, it is expressed as:
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W **qMqg0T (2.28)
* Where q̇i (t) is a virtual velocity vector containing a set of fictional velocities that verify the velocity constraint equations, given by Equation (2.7), considering the right hand side vectors of the velocities null. The term Mq represents the inertial forces, in which M is the global mass matrix of the system and q is the vector of accelerations, and g represents the vector of forces, containing the externally applied forces and moments as well as the quadratic velocity terms associated to inertial forces.
The global mass matrix M encloses the mass matrices of every individual body Mi of the mechanical system. In this work, the local reference frame of each body is considered to be aligned with the respective principal inertia frame, making the mass matrix of each body diagonal:
MMMMMiiiindiag[m ,m ,m ,J ,J ,J]diag[,,,] 12 (2.29) where mi is the mass and Jξ , Jη , and Jζ are the principal moments of inertia of body i and n is the number of bodies. Note that if the local reference frame of the rigid body positioned at the centre of mass is not aligned with the principal axes of inertia, the mass matrix of the body is not diagonal, since other terms, products of inertia, will appear. Moreover, if the body fixed coordinate frame does not have its origin in the body centre of mass the mass matrix is not diagonal because the 1st moments of inertia are not null. The vector of forces g gathers the individual forces and moments associated to each body:
T f f,, f f i x y z T g f,,,,,, n f n f n (2.30) T 1 1 2 2 nn ninnn,, ω i J i ω i where fi represents the global force vector, n′i denotes the local moment vector and J′i represents local inertia tensor associated to body i. The local moment vector also contains the quadratic velocity terms of body i. It is worth noting that the internal forces of the system are not taken into account in Equation (2.28) since they do not produce virtual power. Nevertheless, these forces can be obtained using the Lagrange multipliers method, which is given by (Nikravesh, 1988):
()cT g Φλq (2.31) where g(c) is a force vector containing the internal constraint forces and λ is the vector of Lagrange multipliers. From the physical point of view, the rows of the Jacobian matrix provide the direction of the constraint forces, while the vector of Lagrange multipliers provide their magnitude. Since the internal forces do not produce virtual power, they can be added to Equation (2.28) without infringing the principle of virtual power. Accordingly, Equation (2.28) is rewritten as:
**TT W q Mq g Φq λ 0 (2.32)
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* Regardless of the instant of time, there is always a solution for the virtual velocity vector q̇i and vector of Lagrange multipliers λ that makes the term in parenthesis null. This is the term that represents the equations of motion of a constrained multibody system, which is expressed in its final form as:
T Mq Φq λ g (2.33)
It must be noted, at this time, that if Equation (2.33) needs to be solved in a forward dynamic analysis, the constraint acceleration equation set, Equation (2.9), must be used simultaneously to assure that the number of unknowns equals the number of equations.
2.5 Inverse Dynamic Analysis
The inverse dynamic analysis is a procedure for calculating the internal and external forces acting in the system and by the system, according to its topology, kinematic constraints and pre-defined motion (Silva, 2003). The application to biomechanical systems allows to obtain the joint torques and muscle forces developed by the locomotion apparatus of a biomechanical model. In order to perform an inverse dynamic analysis of a mechanical system, it is required to know the motion of the task under analysis in advance and check if it is kinematically consistent with the system, i.e., if the kinematic constraint equations given by Equation (2.3) are fulfilled. After performing a kinematic analysis to ensure the consistency of the positions, velocities and accelerations, with the biomechanical model to be used, the next phase includes the assemblage of the global mass matrix and force vector based on the description of the mechanical system and the kinetic data recorded during the task under analysis, respectively. The externally applied forces measured are introduced into the force vector. The equations of motion that govern the mechanical system can therefore be assembled and rearranged:
T Φq λ g Mq (2.34)
Generally vector g includes known external forces gk , such as the ground reaction forces, and unknown external or internal forces gu , such as joint torques or muscle forces. If these unknown forces are formulated as applied forces fext:
gCfuext (2.35) where C is a matrix that appropriately places the forces and their corresponding transport moments in each body. The inverse dynamics equation is now written as:
λ T Φq C gk Mq (2.36) fext in which the unknown Lagrange multipliers and forces are unknowns.
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In this work the unknown joint torques and muscle forces are obtained by imposing kinematic constraints. The Lagrange multiplier vector contains these forces and, therefore, Equation (2.34) is used to solve the inverse dynamic problem. The Lagrange multipliers method states that with each external force is related a kinematic driver responsible for guiding a specific DOF (Silva, 2003), being introduced in the Jacobian matrix by the driver constraint equations, which are equal in number to the DOF of the mechanical system. The equations of motion, given by Equation (2.34), are solved with respect to the Lagrange multipliers vector to obtain the internal forces, related to the kinematic constraints, and the external forces, related to the driving constraints. Although the methodology to perform an inverse dynamic analysis of determinate mechanical systems was presented, this work also deals with indeterminate mechanical systems. In this case there are more constraints, caused by the introduction of muscle actuators in the lower limbs, resulting in more unknowns than equations of motion. This problem is solved by using optimization procedures and is explained in detail in Chapter 5.
2.6 Joint Torques
Due to its importance in the inverse dynamic analysis of the multibody biomechanical model, the explicit calculations of the joint torques, implied by Equation (2.31) are detailed here. The joint torque for an anatomical joint represented by a revolute driver is given by:
0 f (sd ,1) i T (sd ,1) v uAi ni (rdrd ,1)( Trd ,1)( ,1) g q (2.37) 0 f (sd ,1) j T (sd ,1) u vA j nj
(,1)rd (,1)rd where ni and nj are the joint torques associated to body i and j, respectively. Similarly, the joint reaction forces associated to the spherical driver constraint are the joint torques obtained by using Equation (2.31) once more, which are explicitly written as:
000 f (sd ,3) i TTT (sd ,3) (sd ,3) uAAui j uAAu i j uAAu i j n i j i j i j 1 i (,3)(,3)sd sd T (,3) sd g q 2 (2.38) 0 0 0 (sd ,3) f (sd ,3) 3 j TTT (sd ,3) uAAuj i uAAu j i uAAu j i nj j i j i j i
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In Equation (2.38) it is clear that the spherical driver constraint only applies intersegmental torques between bodies connected by a spherical joint, as expected. Note that for any rotational driver considered, the joint reaction forces fi and f j associated to each body are null. The components of such
( ,3)sd ( ,3)sd joint torques, ni and nj , are expressed in bodies i and j coordinate systems, respectively, as it always occurs when using a formulation based in Cartesian coordinates, as the one used here.
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3 Biomechanical Model of the Human Skeletal System
A biomechanical model is a mathematical discretization of the highly complex human body system. The level of detail of the model should be adapted to fulfil the requirements of the motion under analysis. Several different types of applications use biomechanical models to predict the mechanical behaviour of the human body during the performance of various activities. In this work, a full body representation was considered to perform a normal gait pattern analysis. The anthropometric model used is described along with the dimensional and physical properties of each anatomical segment.
3.1 The Anthropometric Model
An anthropometric model is considered a segmented representation of the body geometry including essential data concerning the dimensional and physical properties of the anatomical segments (Seireg & Arkivar, 1989). The anthropometric model applied in this work is based on the combination of two models: a computer simulation code SOM-LA (Laananen, 1991; Laananen et al., 1983) and a general- purpose biomechanical model (Celigüeta, 1996). The latter anthropometric model provides data for the feet, hands and head of the human body, while the rest of the human body is described by the former model. The anthropometric data is obtained in the work of Chandler, et al. (1981), considering the mass distribution and body size of the 50th percentile male. The anthropometric model and data were compiled by Silva (2003) and are reproduced here. The model divides the human body in sixteen anatomical segments, as illustrated in Figure 3.1.
ID Rigid Bodies Description 16 1 Right Foot From ankle to toe. 15 2 Right Lower Leg From knee to ankle. 3 Right Upper Leg From hip to knee. 11 10 12 4 Lower Torso From the first lumbar vertebrae to the bony pelvis. 5 Left Upper Leg From hip to knee. 9 13 6 Left Lower Leg From knee to ankle. 7 Left Foot From ankle to toe. 8 4 14 8 Right Hand From wrist to finger tips. 3 5 9 Right Lower Arm From elbow to wrist. 10 Right Upper Arm From shoulder to elbow. 11 Upper Torso From the first thoracic vertebrae to the twelfth. 12 Left Upper Arm From shoulder to elbow. 2 6 13 Left Lower Arm From elbow to wrist. 14 Left Hand From wrist to finger tips. 15 Neck From the first cervical vertebrae to the seventh. 1 7 16 Head Cranium, upper and lower jaws. Figure 3.1: Anthropometric model description. Illustration and table adapted from Silva (2003).
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3.2 Body Dimensions and Properties
The sixteen anatomical segments dimensions and physical properties based on the 50th percentile male, required to define the biomechanical model, are presented here. Figure 3.2 illustrates the anthropometric model in a standing position with the local reference frames and annotations referring to the lengths of each anatomical segment. The anthropometric data concerning the mass, length and inertia, as well as the initial position, of each anatomical segment are compiled in Table 3.1.
16 16 16 15 15 16 L 15 16 d16 16 15 10 15 L d 15 10 16 d15 11 12 12 d10,12 10 11 L16 (c) d L10,12 9,13 11 L 9 12 9,13 9 (b) L11 13 13 9 4 4 13 8 11 8 4 14 L11 8,14 14 d 11 8,14 8,14 8 3 L8,14 d11 5 14 3 5 d3,5
3 L3,5 5 L8,14 L4
(d) 2 d4 6 2 d 2,6 L4 6
2 Y 6 1,7 (e) L2,6 d1,7 Z 1 L1,7 1,7 1 7 d1,7
7 L1,7 1 X 7 (f) (a)
Figure 3.2: Representation of the anthropometric model. The annotations denote the body-fixed frames, the major dimensions and location of the centre of mass of each anatomical segment. (a) Perspective view in the standing position; (b), (c) Sagittal view of the head and neck; (d) Frontal view of the hand; (e) Frontal view of the upper and lower torso; (f) Sagittal view of the foot.
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Regarding the physical characteristics, namely, the inertia, the local reference frame of each anatomical segment rigidly attached to its centre of mass is assumed to be aligned with the principal inertia axes of each segment. Accordingly, the moments of inertia calculated with respect to the axes of the local reference frame correspond to the principal moments of inertia.
Table 3.1: Anthropometric data of the sixteen anatomical segments. The lengths, Li and L̲i , and the distance to the centres of mass, di and d̲i , are schematically represented in Figure 3.2 (adapted from Rodrigo et al., (2008)). Principal Moments Mass [Kg] Length [m] CM distance [m] Initial Position [m] -2 2 Nbr. of Inertia [10 Kg.m ] m Li / L̲i di / d̲i x y z Iξξ Iηη Iζζ 1 1.182 0.069 / 0.271 0.035 / 0.091 0 0.034 0.094 0.1289 2.569 0.128 2 3.626 0.439 / - 0.215 / - 0 0.357 0.094 1.086 3.14 3.83 3 9.843 0.434 / - 0.151 / - 0 0.727 0.094 1.435 3.14 15.94 4 14.2 0.275 / 0.188 0.064 / - 0 1.006 0 26.22 26.22 13.45 5 9.843 0.434 / - 0.151 / - 0 0.727 -0.094 1.435 3.14 15.94 6 3.626 0.439 / - 0.215 / - 0 0.357 -0.094 1.086 3.14 3.83 7 1.182 0.069 / 0.271 0.035 / 0.091 0 0.034 -0.094 0.1289 2.569 0.128 8 0.489 0.185 / 0.090 0.093 / - 0 0.873 0.161 0.067 0.148 0.146 9 1.402 0.250 / - 0.123 / - 0 1.093 0.161 0.124 0.298 0.964 10 1.991 0.295 / - 0.153 / - 0 1.358 0.161 1.492 2.487 1.356 11 24.948 0.294 / 0.322 0.101 / - 0 1.318 0 8.625 13.638 21.198 12 1.991 0.295 / - 0.153 / - 0 1.358 -0.161 1.492 2.487 1.356 13 1.402 0.250 / - 0.123 / - 0 1.093 -0.161 0.124 0.298 0.964 14 0.489 0.185 / 0.090 0.093 / - 0 0.873 -0.161 0.067 0.148 0.146 15 1.061 0.122 / - 0.061 / - 0 1.572 0 0.268 0.215 0.215 16 4.241 0.256 / 0.147 0.020 / 0.051 0.051 1.653 0 2.453 2.034 2.2249
3.3 Model Topology
The biomechanical model is a collection of rigid bodies interconnected by kinematic joints, in which the rigid bodies represent the anatomical segments and the joints represent the articulations. The topology of the model is illustrated in Figure 3.3 and the description of the type of kinematic joints considered is summarized in Table 3.2. Figure 3.3 identifies schematically the forty-one DOF of the biomechanical system, from which thirty-five DOF are related to the relative motion of the anatomical segments at the kinematic joints. The remaining six DOF describe the position and orientation of the lower torso, which is responsible for the motion of the whole body as a unit.
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15 16
15 10 12 11
11 10 12 9 4 13 9 8 13 4 5 8 3 14 14
3 5
2 6
2 6
1 7 1 7
(a) (b)
Figure 3.3: Schematic representation of the sixteen rigid bodies and fifteen kinematic joints. (a) Description of the biomechanical model topology (adapted from Silva, (2003)); (b) Exploded view of the biomechanical model, including the identification of the forty-one DOF.
Table 3.2: Description of the joints used to interconnect the rigid bodies (adapted from Silva, (2003)).
Nbr. Joint Type DOF Joint Centre Location
1, 7 Ankle Spherical 3 Midpoint between the medial and lateral malleolus. Midpoint between the maximal protrusions of the femoral 2, 6 Knee Revolute 1 epicondyles. 3, 4 Hip Spherical 3 At the centre of the femoral head. 5 Back Spherical 3 Between 12th thoracic and 1st lumbar vertebrae Centre of a transverse section of the capitate bone, at the 8, 14 Wrist Universal 2 level of the grove between the lunate and capitate bones. Centre of the transverse section of the humerus at the 9, 13 Elbow Universal 2 level of the medial humeral epicondyle. 10, 11 Shoulder Spherical 3 At the centre of the humeral head. 12 Torso/Neck Spherical 3 Between 7th cervical and 1st thoracic vertebrae. 15 Neck/Head Revolute 1 At the occipital condyles.
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3.4 Scaling Anthropometric Data
The anthropometric model presented corresponds to the standard 50th percentile human male. However, the subjects under analysis can have different physical characteristics from the standard model, namely, regarding the length, weight and moments of inertia of the anatomical segments. Therefore, a scaling procedure is needed to match the characteristics of the anthropometric model with those of the subject under analysis. The scaling procedure implemented establishes non-dimensional scaling factors by comparing the data estimated in the laboratory with the equivalent data from the 50th percentile human male (Silva, 2003). These scaling factors were defined by Laananen (1991) as:
Lm ii 2 (3.1) LimiIimiLi5050thth,, Lmii
where Li , mi and Ii are respectively the scaling factors of the length, mass and moments of inertia calculated for segment i. The length of each individual anatomical segment is calculated for the subject under analysis throughout one gait cycle and the resultant average length, Li, is compared to the length
50th of the anatomical segments of the reference model, Li . Since the mass of each segment cannot be measured in the laboratory, the mass ratio is calculated by comparing the total mass of the subject, mi,
50th with the total mass of the reference model, mi . Once these two scaling factors, of length and mass are known, the inertia scaling factor is easily calculated according to Equation (3.1). The scaled anthropometric model is obtained by multiplying the biomechanical parameters of the reference model with the corresponding scaling factors. It should be noted that the scaling procedure, suggested here, is only applicable to male subjects whose percentile is not too different from that of the 50th percentile. For individuals of different gender, very large or very small percentiles or with pathologies at the level of the anatomical segments caution must be exercised when using these scaling factors.
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4 Inverse Dynamic Analysis of Determinate Biomechanical Systems
The human motion can be investigated using a mechanical approach of the biomechanical system to calculate the reaction forces and net moments of force developed at the joints of a subject under analysis. The articulations are represented by kinematic joints, which are responsible for simulating their relative motion, and the muscle actions are represented by rotational drivers, which are responsible for generating moments of force at the articulations that replace the muscle dynamic actuation. This approximation results in the creation of a determinate biomechanical system, i.e., the number of unknowns is equal to the number of unknowns associated to the joint torques. An inverse dynamic analysis of a complex human activity usually requires the acquisition of important input data, which can be divided into three distinct types of information: anthropometric information about the subject, i.e., total body mass and height; kinematic information, characterised by a set of trajectories of markers placed on relevant palpable bony landmarks, which is used to describe the motion of the anatomical segments; and kinetic information concerning all external forces applied on anatomical segments, including their application points (Silva, 2003). This information is gathered experimentally, in this work, for a normal gait pattern at the Lisbon Biomechanics Laboratory (LBL). This chapter addresses the processing and filtering of the data acquired at the LBL, the application of a kinematic consistency procedure to ensure that the kinematic data is consistent with the anthropometric data and the biomechanical model implemented to be used in the inverse dynamic analysis of the torque- driven biomechanical system.
4.1 Data Acquisition
The input data collected to perform an inverse dynamic analysis of the human gait is described in this section. Each type of input data is prone to errors and uncertainties, being the kinematic data the most sensitive. For instance, a disturbance in the application points of the ground reaction forces over the feet or a spatial misconstruction of the anatomical segments strongly affects the quality of the results (Silva and Ambrósio, 2004). A filtering technique is also applied to exclude high frequency noise associated to the kinematic and kinetic data while the important information in the acquired data is still preserved.
4.1.1 Anthropometric Data
The gait analysis is performed on a subject exhibiting a normal gait pattern. The subject is a twenty-two years old male with a height of 1.81 m and total body mass of 75 Kg. The height and weight of the subject are measured in order to scale the physical characteristics of the 50th percentile human male biomechanical model, i.e., the mass, length and moments of inertia of each anatomical segment as described in Chapter 3.
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4.1.2 Kinematic Data
The kinematic input data concerns the necessary information to describe the motion of the entire biomechanical system for a gait analysis. The duration of the analysis is defined in this work by one complete gait cycle or stride performed by the subject. The stride is defined as the period of time between an initial event of one foot to the repetition of the same event in the same foot. The initial event is when the right foot makes contact with the ground, which corresponds to the heel contact of the foot with the ground in the case of a normal gait analysis (Winter, 2009). The gait cycle can be divided into two main phases: the stance phase, during which the right foot is on the ground, and the swing phase, where the right foot is no longer contacting the ground and the leg is swinging through to deliver the next heel strike (Vaughan et al., 1999). For a normal gait, the stance phase corresponds to 60% of the gait cycle period while the remaining 40% correspond to the swing phase. There are several different techniques to perform a motion acquisition (Chèze, 2014; Medved, 2001; Cappozo et al., 2005; Silva, 2003). In this work, the motion capture is performed using fourteen synchronized infrared cameras placed around the laboratory, as illustrated in Figure 4.1. These cameras trace and record the positions of every reflective anatomical marker placed on the subject’s skin with a sampling frequency of 100Hz. The gait cycle selected for analysis lasts for 111 frames, which corresponds to a stride period of 1.11s. This stride cadence is within the values reported in the literature for a normal gait pattern (Vaughan et al., 1999).
Plate2 Plate3 Plate1
Figure 4.1: Top view of the LBL using the QUALISYS program. Representation of the set of fourteen infrared cameras used to acquire the trajectories of the forty-four reflective markers placed on the subject’s skin and the three force plates used to acquire the ground reaction forces.
During the acquisition procedure, the subject is asked to walk back and forth to become familiar with the environment so that a normal gait cycle can be captured. Several gait cycles are captured being only one selected for the gait analysis. In the selected gait cycle, the subject walks from the right to the left direction as seen from the top view of the LBL in Figure 4.1, being the left foot already in contact
34 with the force plate one, and the right heel in contact with the 2nd force plate. The gait cycle ends when the same event occurs, i.e., the left foot is over the 3rd force plate and the right heel is initiating contact with the ground. Figure 4.2 illustrates the procedure to obtain the kinematic data required to perform an inverse dynamics analysis, starting with the motion acquisition in the laboratory, followed by the identification of the markers and generation of an input file, and finalizing with the visualization in MATLAB.
(a) (b) (c) Figure 4.2: Processing of the markers trajectory. (a) Male subject with reflective markers during a gait analysis at LBL; (b) Reproduction of the markers on the QUALISYS program; (c) Representation of the subject’s motion on MATLAB.
The kinematic description of the subject’s motion relies on a set of trajectories of anatomical reflective markers placed on the subject’s skin, as illustrated in Figure 4.2(a). These markers are placed near the joints and extremities of the subject to provide the position of the articulations centre and, after processing, the position and spatial orientation of the anatomical segments (Wu et al., 2002, 2005). The marker set protocol used is described in Appendix A. Most of the articulation centres are defined as the midpoint between two markers. However, the hip and glenohumeral joint centres cannot be determined this way due to their location. For these joints, so called functional or predictive methods are applied (Kaiz et al., 2015; Michaud et al., 2016). The functional methods calculate the joint centre via an optimization procedure that provides the mean centre of a spatial rotation (Cappozzo, 1984; Leardini et al., 1999). The subject is recorded performing specific motions, for instance an abduction and adduction of the leg, being the acquired data processed to find the hip joint centre. A predictive method determines the location of a joint centre using anthropometry based regression equations (Harrington et al., 2007; Meskers et al, 1998; Campbell et al, 2009). Although predictive methods are more prone to errors than functional methods, they are simpler to implement and do not require the recording of additional motions. For these reasons, the glenohumeral
35 joint centre and the hip joint centre are estimated using predictive methods. To fully define the spatial orientation of an anatomical segment, i.e., to determine the local axes of each segment, the position of at least three anatomical markers are required, but other segments, such as the head, feet, hands and upper and lower torso, need an additional marker to define their orientation and/or extremities. The procedure applied to obtain the spatial orientation of the anatomical segments follows closely the ISB recommendations (Wu et al., 2002, 2005).
4.1.3 Kinetic Data
The kinetic data, the remaining input needed for an inverse dynamic analysis, comprises the external forces and their point of application over the anatomical segments that describe the subject. Some forces are known in advance, such as the gravitational forces, but the externally applied forces are unknown. However, in the case of a gait analysis, the subject surrounding environment is instrumented with adequate force measuring devices to acquire the external forces and points of application. The LBL is equipped with three force plates AMTI OR6-7, schematically represented in Figure 4.1, which measure the ground reaction forces and their respective points of application on the subject’s feet. The three force plates are synchronized with the infrared cameras and have the same sampling frequency of 100Hz. Similarly to the kinematic data, kinetic data have also high frequency noise due to the analog-to-digital conversion and require the use of filtering techniques. The transmission of the ground reaction forces to the biomechanical system is performed by the contact between the foot and the force plate. Figure 4.3 illustrates the force transmission during the right foot push-off as well as the actualization of the vector of forces of each foot. The ground reaction forces can be described by their three components as presented in Figure 4.4. The top of Figure 4.4 depicts the evolution of one gait cycle, corresponding to 0%, 50% and 100%, i.e., when the right foot, left foot and again right foot contact the ground, respectively. The magnitudes of the ground reaction forces are within the values reported in the literature for a normal cadence gait analysis (Vaughan et al., 1999; Winter, 2009).
Y X ri i f Z i GRF fffootmg GRF r g foot COP i sCOP nfoot s COP f GRF ω J ω
COP
Figure 4.3: Transmission of the ground reaction forces to the biomechanical system.
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150 Right foot 100 Left foot
50
0 Fx [N] Fx
-50
-100
-150 0 20 40 60 80 100 % of Stride
800 Fy [N] Fy 600
400
200 Right foot Left foot 0 0 20 40 60 80 100 % of Stride 50 40
Fz [N] Fz 30 20 10
0 -10 Right foot -20 Left foot -30 0 20 40 60 80 100 % of Stride Figure 4.4: Ground reaction forces. Representation of the three components of the ground reaction forces measured in the right and left foot during a complete gait cycle.
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4.2 Data Filtering
The trajectories of the anatomical markers are recorded by the infrared cameras with a high level of precision. However, due to several factors, such as skin movement artefacts (Leardini et al., 2005), vibration transmitted by the contact of the foot with the ground and occlusion of anatomical markers during the swing of the arms and legs, among others, the kinematic data contain a significant level of noise that must be filtered. The same occurs with the kinetic data given by the force plates. During a gait analysis, the conversion of the input signal given by the interaction between the foot and the ground into a digital signal generates high frequency noise that requires the application of a filtering method. These directly measured variables, the global position of the markers and ground reaction forces, are used to perform the inverse dynamic analysis and obtain the joint torques, reaction forces and muscle forces. Hence, cleaner and smoother signals provide more accurate results (Winter, 2009). There are several filtering methods that can be used to reduce the noise associated to the experimental data. The most commonly used is a 2nd order Butterworth low pass filter with a zero phase lag (Winter, 2009). This filter attenuates the noise above a defined cut-off frequency without affecting the frequency below the threshold. When performing a gait analysis, the cut-off frequency usually ranges from 2 to 6 Hz, depending on the stride cadence and on the location of the anatomical markers (Silva, 2003). The anatomical markers placed in the extremities of the limbs exhibit larger position variation than the ones placed, for instance, on the torso. Consequently, the former anatomical markers require the use of higher cut-off frequencies than the latter ones (Silva, 2003). The selection of the appropriate cut-off frequency is important because if the cut-off frequency is too high, undesired noise can be present in the filtered signal and if the cut-off frequency is too low, important features of the signal can be absent in the filtered signal (Silva, 2003). There are several methods to help choosing the best cut-off frequency, but these are not discussed in this work. The interested reader is referred to the work of Winter (2009) for further details. In this work, a cut-off frequency of 4 Hz is applied to all anatomical markers and a cut-off frequency of 10Hz is applied to the ground reaction forces measured by the force plates. Figure 4.5 schematically depicts the application of a 2nd order Butterworth low pass filter.
Low pass filter X X XXF F
Signal Noise Signal Noise
Frequency fc Frequency Frequency (a) (b) (c) Figure 4.5: 2nd order Butterworth low pass filter. (a) unfiltered signal; (b) low pass filter representation; (c) filtered signal. (adapted from Winter (2009)).
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The filter is implemented using the MATLAB function butter, which allows the configuration of the Butterworth filter order and the cut-off frequency. However, when a Butterworth filter is applied to an unfiltered signal, a phase distortion is inserted in the neighbourhood of the cut-off frequency. To correct this unwanted phase-lag, the once-filtered data is filtered again in the reverse direction of time. This procedure doubles the order of the first applied filter, resulting in the creation of a fourth order zero-phase-shift filter (Winter, 2009). Once more, a MATLAB function, named filtfilt, was used to filter the signal and correct the phase distortion.
4.3 Consistent Kinematic Data
The kinematic data acquired do not usually respect the kinematic constraints associated to the biomechanical model, resulting in constraint violations that affect the joint reaction forces and torques generated by the solution of an inverse dynamic analysis (Silva and Ambrósio, 2002). During the process of motion acquisition there are several sources of errors, such as the misplacement of anatomical markers (Della Croce et al., 2005), the scaling procedure applied to match the anthropometric dimensions of the subject under analysis and the enforcement of ideal mechanical joints where in fact the human articulations are imperfect joints. These errors lead to non-consistent positions and orientations of the anatomical segments with the kinematic structure of the biomechanical model. To impose the consistency of the kinematic data with the biomechanical system constraints, the kinematic positions are modified in order to satisfy the constraint equations. The procedure followed to calculate the consistent positions consists in: (1) determining, from the initially irregular positions of the anatomical markers obtained by the motion acquisition, the average anatomical segment lengths, which by applying the length scaling factor described in Section 3.4, provide the constant scaled segment dimensions of the biomechanical model; (2) defining the orientation, i.e. the local reference frame of each segment, saved as Euler parameters; (3) describing the time dependent joint angles as well as the position and orientation of the base body (lower torso) that are responsible for driving the DOF of the system; (4) performing a kinematic analysis to obtain the positions that fulfil the constraints of the biomechanical model. The velocity and acceleration of the joint angles and the base body are calculated using direct spline differentiation. The kinematic consistent velocities and accelerations of the system are obtained by solving the Equations (2.7) and (2.8), as described previously in Chapter 2. To verify if the kinematic consistent positions describe the motion captured in the LBL, the graphical tool SAGA is used to convert the consistent vector of coordinates into an animation. Figure 4.6 depicts the evolution of one gait cycle, corresponding to 0%, 60% and 100%, as well as the frontal view of the skeleton model using the graphical tool SAGA. The application of this animation tool, is very useful since it allows to detect obvious problems in the positions and orientations of the anatomical segments that are, otherwise, concealed in the vector of coordinates. However, note that visual
39 inspection of the graphical display does not guarantee the accuracy of the models and results but simply helps identifying obvious problems, while providing an insight to the biomechanical model behaviour.
(a) (b) Figure 4.6: One gait cycle illustrated by the skeleton model of SAGA a) Sagittal view. right heel contact at 0%, right toe-off at 60% and again right heel contact at 100%; b) frontal view.
4.4 Application Case to a Normal Gait Cycle
The solution of the determinate inverse dynamic analysis of the human gait acquired provides the joint reaction forces and torques developed by the driver actuators at the joints. These are calculated by solving the equations of motion of the system with respect to the vector of Lagrange multipliers, as described in Section 2.5. Figure 4.7 presents the global components of the joint reaction forces in the right ankle, knee and hip and Figure 4.8 depicts the joint torques developed at the same joints in the sagittal plane. Both results are scaled to the total body mass of the subject. Note that in the top of these figures, and hereafter, it is illustrated the evolution of one gait cycle, corresponding to 0%, 60% and 100%, i.e., right foot initial contact, right toe off and second contact of the right foot, respectively. The joint reaction forces displayed in Figure 4.7 for the right leg joints are symmetric to the ground reaction forces presented in Figure 4.4, as expected. The X component of the joint reaction forces in Figure 4.7(a) shows two peaks, a positive and a negative in magnitude, that correspond to the initial contact of the right heel and the right toe off, respectively. As for the Y component in Figure 4.7(b), which is the largest of the three components, has two peaks during the stance phase, i.e., after the left foot take off and before the same foot contacts with the ground. Figure 4.7(c) depicts the evolution of the Z component, which as an inward direction, pointing from the centre of the pelvis to the ground. These results are in agreement with the literature (Silva, 2003; Winter, 2009) being characteristic for normal gait patterns.
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1.5
1
0.5
0
-0.5
Force/Bodymass[N/kg] -1 Ankle -1.5 Knee Hip -2 0 20 40 60 80 100 % of Stride (a) 4
2 0 -2
-4 -6
Force/Bodymass[N/kg] -8 Ankle -10 Knee Hip -12 0 20 40 60 80 100 % of Stride (b) 0.4
0.2
0
Force/Bodymass[N] -0.2 Ankle Knee Hip -0.4 0 20 40 60 80 100 % of Stride (c) Figure 4.7: Reaction forces in the right ankle, knee and hip joints obtained during one gait cycle analysis. a) X component. b) Y component. c) Z component.
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2 Winter (2009) 1.5 Silva (2003) + Extension 1 Current model
0.5
0
-0.5
-1 Moment/Bodymass[N.m/kg]
-1.5 0 20 40 60 80 100 % of Stride (a) 1 Winter (2009) Silva (2003) + Extension 0.5 Current model
0
-0.5 Moment/Bodymass[N.m/kg]
0 20 40 60 80 100 % of Stride (b) 2 Winter (2009) 1.5 Silva (2003) + Plantarflexion Current model
1
0.5
0 Moment/Bodymass[N.m/kg]
-0.5 0 20 40 60 80 100 % of Stride (c) Figure 4.8: Joint torques developed in the sagittal plane at the right leg. The results obtained by Silva (2003) and Winter (2009) are also presented for comparison. a) hip b) knee and c) ankle.
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The joint torque at the hip should be positive (extension) until the middle of the stance phase of the right leg to control the knee flexion and then negative to bring the thigh forward after the left heel contacts with the ground (Winter, 2009). The knee should produce a negative joint torque (flexion) to support the contact of the right heel, a positive moment when the hip is aligned with the knee, then decreases to control the left leg swing becoming positive after the left heel touches the ground. The joint torque at the ankle starts to be negative (dorsiflexion) during the contact of the plantar surface of the foot with the ground and evolves to a growing positive moment reaching a peak at the right toe off. Although the moment at the ankle joint compares well with the results of the literature, the moments at knee and hip joints reveal differences. Such differences can be justified by the possible inaccuracies in the calculation of the hip joint centre (Leardini et al., 1999), which affects the hip and knee moment. In the work of Stagni et.al (2000), a sensitivity analysis was performed to verify the effects of the hip joint centre mislocation on gait results. The authors concluded that a 30 mm anterior or posterior mislocation of the hip joint centre resulted in a propagated error in the sagittal moment of about 22% and a delay of the flexion-to-extension timing of about 25%, respectively, thus justifying the differences in the evolution of the joint torque patterns described here. A note should be made concerning the calculation of the joint torques by the authors presented in comparison with the current model. The work of Silva (2003) uses a three-dimensional biomechanical model described using a multibody formulation with full Cartesian coordinates, which closely relates to the formulation used here. On the other hand, Winter (2009) uses a two-dimensional model and the Newton-Euler three-dimensional equations of motion to evaluate the moments, segment by segment, i.e., using free body diagrams, following the kinematic chain from the foot to the hip. The formulation used by Winter does not depend on the constraint equations being also implemented in this work to provide an independent confirmation of the results of the multibody formulation.
4.5 Validation of Results Using the Classical Newton-Euler Method
In order to provide an independent indication on the appropriate implementation of the multibody methodology, and thus in the results presented in the Section 4.4, the solution obtained is compared with that obtained by using the classical Newton-Euler equations. The classical Newton-Euler methodology is used to solve the dynamic equilibrium equations of each rigid body of an open-loop system sequentially, starting from the most distal body and moving inwards along the kinematic chain (Silva, 2003). This method is only applied to open loop structures, which is the case of the biomechanical model developed here, since for closed loop structures the equations become redundant. Therefore, the classical Newton-Euler procedure needs to be customised to the kinematic structure of the biomechanical model and can only be applied if, throughout the motion under analysis, there are no closed loops. In contrast, the multibody methodology can be applied to biomechanical models with open and close loops.
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However, the solution provided is obtained for all anatomical segments at the same time, leading to a possible error propagation from one kinematic chain to another (Silva, 2003). The analytical model used to perform the calculations of the reaction forces and moments at the joints is defined only for the right lower limb, which aggregates three anatomical segments, foot leg and thigh, interconnected, by the ankle, knee and hip joints. For the results to be equivalent, both methodologies share the same anthropometric, kinematic and kinetic data. The Newton-Euler equations of motion for an unconstrained spatial rigid body are written as (Winter, 2009):
faammG (4.1) nh G where ∑ f is the sum of all the externally applied forces, m is the mass of the rigid body, a is the linear acceleration vector of the centre-of-mass, aG is the gravitational acceleration vector, ∑ n is the sum of all externally applied moments-of-force, and ḣG is the first time derivative of the angular moment vector relative to the centre of mass of the rigid body. For an unconstrained spatial rigid body with the orientation of the local reference frame coincident with its principal axes of inertia the vector ḣG , is written as (Winter, 2009):
hIII () hIII () (4.2) hIII ()
where Iξ , Iη and Iζ are the principal moments-of-inertia and ωξ , ωη and ωζ and , and are the angular velocity and acceleration of the centre of mass, respectively. The free-body diagrams of the foot, knee and hip are illustrated in Figure 4.9. The equations of motion of these anatomical segments can be easily deduced an are expressed by Equations (4.3-4.5).
nhip nankle fankle fknee
fhip n knee fknee nankle
f f floor ankle n knee (a) (b) (c) Figure 4.9: Free-body diagrams of the rigid bodies describing the a) foot, b) leg and c) thigh adapted. The unknowns are the forces and moments located in the proximal joint.
44
faafankleGfloorm() (4.3) nsfsfhanklefflooraankleG
faafkneeGanklem() (4.4) nsfsfnhkneekkneeaankleankleG
faafhipGkneem() (4.5) nsfsfnhhiphhipkkneekneeG
The order in which the equations are solved follows the kinematic chain, i.e., first the foot, then the knee and finally the hip joint. Starting with the foot, the solution of Equation (4.3) provides the joint reaction force at the ankle fankle and the moment-of-force nankle , which are the unknowns of Equation
(4.4). Accordingly, the joint reaction force at the knee fknee and the moment-of-force nknee can be calculated. The final unknowns, related to the thigh, are obtained similarly. The solution of Equation
(4.5) provides the joint reaction force at the hip fhip and the moment-of-force nhip . The moments-of-force in the sagittal plane developed at the right leg, specifically at the ankle, knee and hip joints, obtained with both methodologies, multibody and analytical, are compared in Figures 4.10 and 4.11.
1 Multibody Analytical 0.5
0
-0.5 Moment/Bodymass[N.m/kg]
-1 0 20 40 60 80 100 % of Stride
Figure 4.10: Moment of force developed by in the sagittal plane at the right hip. The results were obtained using the multibody and analytical methods.
45
1 Multibody Analytical 0.5
0
-0.5 Moment/Bodymass[N.m/kg]
-1 0 20 40 60 80 100 % of Stride (a) 2 Multibody Analytical 1.5
1
0.5
Moment/Bodymass[N.m/kg] 0
0 20 40 60 80 100 % of Stride (b) Figure 4.11: Moment of force developed in the sagittal plane at the right leg: a) knee and b) ankle. The results were obtained using the multibody and analytical methods.
The results obtained for the right lower limb are coincident, regardless of the methodology applied. In other words, the multibody methodology leads to the same results as the analytical method, which shows that the numerical methods used to solve the problem are properly defined and implemented.
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5 Inverse Dynamic Analysis of Indeterminate Biomechanical Systems
In the previous chapter, a biomechanical system of the whole body was applied to calculate the internal forces developed during a normal gait cycle. Drivers actuators were considered to replace the detailed muscle apparatus, which resulted in the formulation of a determinate problem since there is a unique solution to the inverse dynamic analysis. However, this analysis relies on the assumption that the lower muscle apparatus is simulated by rotational drivers located at each joint, which corresponds to a gross approximation and simplification of the reality being unsuitable for important biomedical applications. In order to perform a more faithful analysis of the human biomechanics, the rotational drivers are substituted by muscle actuators, which represent the complex human musculoskeletal system. The introduction of these muscle actuators into the biomechanical model transforms the deterministic solution of the inverse dynamic analysis into an indeterminate inverse dynamic problem, known in the literature as the redundant muscle force sharing problem in biomechanics (Yamaguchi et al., 1995), since the number of muscles at a joint exceeds the number of equations of motion. Consequently, a unique solution cannot be obtained. From a physiological perspective, the indeterminacy introduced by the muscle actuators results from the fact that different muscle activation patterns can generate the same posture or movement of the human body and it is the central nervous system responsibility the selection and activation of the most appropriate set of muscles to perform a human activity (Silva, 2003). Computationally, this problem is generally solved through optimization procedures. The objective of the optimization procedures is to minimize physiological criteria, which represents the decisions taken by the central nervous system, and to obtain an estimation of the muscle forces of the subject under analysis. The optimal solution must also satisfy the equations of motion. The steps taken to solve the indeterminate biomechanical problem and to obtain the muscle forces developed by the lower extremity are presented in this chapter. The first step concerns the multibody formulation of the muscle actuators and the description of the Hill-type muscle model implemented to simulate the muscle contraction dynamics (Zajac, 1989). The input data, anthropometric, kinematic and kinetic data, concerning the subject under analysis, are inherited from the determinate analysis addressed in Chapter 4. The second step is the application of a scaling method to adjust the lower extremity muscle apparatus properties from the work of Yamaguchi (2001) to the anthropometric characteristics of the subject under analysis. The third and final step is the application of an optimization procedure to calculate the muscle activations and, subsequently, the muscle forces. From the several different performance criteria (Silva, 2003), the two most commonly objective functions applied are considered here (Collins, 1995; Crownshield and Brand, 1981). Note that the performance criteria considered here can be handled in a frame by frame basis, which is generally designated as static optimization. Other
47 performance criteria, such as the metabolic energy (Ackermann, 2007), require the solution of the complete gait cycle at the same time for which global optimization is needed. Therefore, performance criteria that involve time are not considered here.
5.1 Muscle Actuators in Multibody Systems
The skeletal muscles are introduced in the biomechanical system as kinematic driver actuators, being each muscle associated to a constraint equation (Silva and Ambrósio, 2003). These kinematic drivers are called muscle actuators to differentiate them from the kinematic drivers used to obtain the joint torques. Depending on the complexity of the path of each muscle, muscle actuators can be described by two or more points. The starting and ending points correspond to the origin and insertion of the muscle, respectively. Although, some muscles require more points, called via points, to simulate the muscle wrap around joints as well as the interaction between two contacting muscles. Figure 5.1. depicts two skeletal muscles of the lower extremity muscle apparatus to schematically represent two different muscle actuator definitions, i.e., with and without via points.
TSL O TSLO
S m1 Tensor O S TSLV1 O Tensor Fasciae TSLV1 Latae Semimembranosus Fasciae Latae
m2
TSLV 2
SI TSLV 2 SI m3 TSLI
TSLI
Figure 5.1: Muscle actuators with and without via points. The semimembranosus, defined as a two- point muscle actuator and the tensor fasciae latae, defined as a four-points muscle actuator due to the muscle path complexity (adapted from Silva and Ambrósio, (2003)).
The muscle length changes during the analysis are described by a constraint equation associated to each muscle actuator. These constraint equations allow the distance between two points to change according to a specific length change history previously determined (Silva and Ambrósio, 2003). The formulation of the constraint equation is demonstrated for muscles modelled with only two points, such as the semimembranosus. Considering the two-point actuator illustrated in Figure 5.2, for which the muscle path is modelled as a straight line with its origin O located in body i and its insertion I located in body j, the constraint equation is written as:
48
1 (,1)mT2 Φdd Ltm ()0 (5.1) where Lm(t) is the total muscle length, which varies with respect to time, and d is the global vector between the origin and the insertion of the muscle, given by:
IO d rj A j s j r i A i s i (5.2)
In Equation (5.2), ri and rj , and Ai and Aj are, respectively, the position vectors and the transformation
O O matrices of bodies i and j. The vectors si′ and sj′ are the local coordinates of the origin and insertion points in body i and j, respectively.
(i)
O si
i i Y ri
i X
j j rj Z
j I s j (j)
Figure 5.2: Muscle actuator defined between its origin O and insertion I.
The contribution to the Jacobian matrix is deducted by differentiating Equation (5.1), which results in the following expression (Quental, 2013):
r11ω ri ω i r j ω j r n ω n 1 TTTT Φ(m ,1) 00 dGsLd O T dGsLd O T 00 (5.3) q 1 i i i j j j ddT 2 where G is a global transformation matrix and s̅′ is a skew-symmetric matrix defined as:
0 sx s y s z e e e e 1 0 3 2 s 0 s s Gs e e e e and x z y (5.4) 2 3 0 1 s s 0 s e e e e y z x 3 2 1 0 sz s y s x 0
The lines of the Jacobian matrix in Equation (5.3) express the direction of the force produced by a two- point muscle actuator and the corresponding Lagrange multiplier represents the magnitude of this force.
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The muscles with more complex paths, which require more than two points to be defined, are introduced in the Jacobian matrix as a sum of several two-point muscle actuators. This procedure is explained for the muscle tensor fasciae latae, illustrated in Figure 5.1, which can be divided into three straight line muscles, named m1, m2 and m3. The insertion point of muscle m1 is coincident with the origin of muscle m1 and the insertion point of muscle m2 is coincident with the origin of muscle m3.
Considering the muscle actuators m1, m2 and m3 associated to the Lagrange multipliers λm1, λm2 and λm3,
T respectively, the contribution to the term of internal forces Φq λ of the equations of motion is:
m m m 1 2 3 000 q Φq00m1 iim1 mmm123 q jjjjΦqΦqΦq m2 (5.5) m3 q00kkΦqm3 000 where qi , qj and qk indicate the rows of the Jacobian matrix and represent the vector of coordinates of the rigid bodies i, j and k, respectively. Considering that any muscle presents, from its origin to its insertion, a constant force, then all straight line muscles, representing the complex muscle path must produce equal forces. As a result, the Lagrange multipliers associated to each muscle element of the tensor fasciae latae form the relation:
mmmTFL123 (5.6) and, accordingly, Equation (5.5) can be simplified to:
mmm 123 0 q Φqm1 ii mmm123 q jjjjΦqΦqΦq TFL (5.7) m3 qkkΦq 0
Equation (5.7) shows that the contribution of more complex muscles to the equations of motion can be determined by segmenting them into several straight line muscles and adding up their individual contribution. Furthermore, regardless of the number of points that define a muscle, each muscle has only one associated kinematic constraint and corresponding Lagrange multiplier. To simplify the biomechanical model and reduce the computational cost, the muscle actuators are only applied in the right leg. Accordingly, since the right leg is driven by the muscle actuators, the rotational drivers considered in the determinate problem are removed, namely, the hip, knee and ankle joint drivers. The remaining anatomical segments, for which no muscle actuators are considered,
50 continue to be actuated by rotational drivers. Therefore, the Jacobian matrix needs to be rewritten to remove the rotational drivers associated to the right leg and add the contributions of each muscle actuator of the lower muscle apparatus. The inverse dynamic analysis gives an integrated solution in terms of the muscle and joint reaction forces, i.e., muscle forces are accounted for during the calculation of the reaction forces at the joints of the biomechanical model and vice-versa (Silva and Ambrósio, 2003).
5.2 Dynamics of Muscle Tissue
The dynamics of muscle tissue can be divided into activation and muscle contraction dynamics (Zajac, 1989), where a neural signal is transformed into the activation of a contractile muscle, which in turn is transformed into muscle force, as schematically represented in Figure 5.3.
Neural signal Activation Muscle activation Muscle Contraction Muscle force Dynamics Dynamics
Figure 5.3: Schematic representation of the dynamics of muscle tissue (based on Zajac, (1989)).
The activation dynamics describes the time lag between a neural signal and the corresponding muscle activation while the muscle contraction dynamics corresponds to the conversion of muscle activation into muscle force (Zajac, 1989). The simulation of this transformation process is based on mathematical models of the muscle and soft tissue that relate the muscular force produced by a muscle with the resulting movement produced in the musculoskeletal system (Yamaguchi, 2001). From all the several mathematical models used to simulate the muscle behavior, the Hill muscle model, and respective variations, are the most commonly used by researchers to simulate both passive and active muscle action (Yamaguchi, 2001). In this work a Hill-type muscle model is implemented to simulate the muscle contraction dynamics, as illustrated in Figure 5.4. The activation dynamics is neglected.
Contractile Element (CE) Total muscle force FtM ()
L Rigid tendon M Passive Total muscle M Element (PE) force FtM () Pennation
LT angle
Figure 5.4: Hill type muscle model used to simulate muscle contraction dynamics. The damping element is included in the contractile element (CE) and the series elastic element (SEE) is neglected.
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The muscle model applied is composed by a contractive element (CE) and a passive element (PE), both of which contribute to the generation of the total muscle force Fm (t). The muscle series elastic element (SEE), which was not represented, is not accounted for since it can be neglected in coordination studies not involving short-tendon actuators (Zajac, 1989). The tendon element is considered rigid (Millard et al. 2013; Oliveira et al., 2015). The contractile and passive properties of the Hill-type muscle model are controlled by two different expressions. The force produced by the contractile element is a function of the current muscle length Lm (t) , rate of length change L̇ m (t) and activation am (t), expressed as:
mmmm mmmmm FLtFLtL (( ))(( ))L FatLtLtatCE ( ),( ),( )( ) m (5.8) F0
m mm mm where F0 is the maximum isometric force, and FL t( ( ) ) and FL t( ( ) ) represent the muscle force- length and force-velocity relationships, respectively (Zajac, 1989). These functions state that the maximum contraction force that a skeletal muscle is capable of producing is dependent on its actual length and on its rate of length change, also named shortening velocity. The analytical expressions for the force-length and force-velocity relationships proposed by Kaplan (2000), based on approximations to experimental, are expressed as:
4 2 9(L )mm 191( tL t) 19 420420LLmm mmm 00 FLtFL (( )) e 0 (5.9)
mm 0L L ( t ) 0 F m Ltm () m m 0 m m m m (5.10) FL ( L ( t )) arctan 5m F0 0.2 L 0 L ( t ) L 0 arctan(5) Lt0 () F m 0 m m m F00 L( t ) 0.2 L arctan(5)
m m where L0 is the optimal muscle fibre length and L0 is the maximum contractile velocity above which the muscle cannot produce force (Zajac, 1989). The maximum contractile velocity is usually between
m m m 10L0 and 15L0 (Yamaguchi, 1995) and, in this work, is considered to be equal to 15L0 . The force produced by the passive element depends only on the current muscle length Lm (t) and is expressed as (Kaplan, 2000):
m 0L0 Lm ( t ) m F 3 Fm( L m ( t )) 80 L m L m 1.63 L m L L m PE m3 0 0 m 0 (5.11) L0 mm 2FLL00m 1.63
52
The expressions presented in Equations (5.9), (5.10) and (5.11), are illustrated in Figure 5.5, to show their behaviour when the muscle is fully activated. These curves represent general isometric and shortening-lengthening contractions performed with different muscle lengths, respectively.
2 2 F F0 Active 0 1.5 Passive 1.5
Total Force[N] 1 Force[N] 1
0.5 0.5
0 0 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5
Length [m] L0 Velocity [m/s] L0
(a) (b) Figure 5.5: Muscle force properties considering a fully activated muscle. a) muscle active and passive force-length relationship. b) muscle force-velocity relationship.
Figure 5.5(a) shows that the contractile element is only able to generate force for muscle lengths between 0 . 5L0 and 1 . 5L0 , and the passive element only produces force when the muscle is stretched above its resting length. The force-velocity relationship, depicted in Figure 5.5(b), shows that the force produced while the muscle is shortening (concentric contraction) is smaller than the force generated during an isometric contraction. On the other hand, when the muscle is stretching (eccentric contraction) the force produced is larger than the maximum isometric force developed by the muscle. Since the passive element is only dependent on the muscle length, the force produced can be easily calculated during the analysis. Hence, it is considered a determined external force, applied at the anatomical segments interconnected by muscles. The active element of the muscle force depends, on the other hand, on the muscle activation, which is unknown. In order to have the Lagrange multipliers associated with the muscles representing muscle activation instead of force, the contribution of each muscle actuator to the Jacobian matrix is multiplied by the following scaling factor (Quental, 2013):
m m m m FL ( L ( t )) FL ( L ( t )) l m (5.12) F0
Considering the muscle activation as design variables, benefits the optimization problem since they are bounded between 0 and 1 (Silva and Ambrósio, 2003).
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5.3 Lower Extremity Muscle Apparatus
The morphological muscle parameters for the modelling of the lower extremity muscle apparatus are presented here. This information is vital for the construction of the muscle actuators and implementation of the muscle model discussed in Sections 5.1 and 5.2, respectively. The muscle database used was initially reported by Delp et al. (1990), was modified by Carhart (2000), and was later reproduced in the work of Yamaguchi (2001). However, the local coordinates that describe the muscles path need to be adjusted since these were defined relatively to a biomechanical model presented in the work of Yamguchi (2001). Figure 5.6. illustrates the lower extremity of the biomechanical model presented in the work of Yamaguchi (2001) with the reference frames as well as the respective anthropometric dimensions. The reference frames attached to the foot, shank and thigh correspond, respectively, to the associated proximal joint, and the reference frame of the lower torso is attached to the midpoint of the line joining the anterior superior iliac spines. The anthropometric dimensions of the 50th percentile male presented are scaled to the dimensions of the Yamaguchi model using the procedure described in Section 3.4. This allows to determine the global position of the centre of mass of the anatomical segments of the lower extremity. Accordingly, the local coordinates of the points defining the muscle path presented in the work of Yamaguchi (2001) are converted to be expressed relatively to the local reference frame attached to the centre of mass.
3 4 4 4 4 4 4 3 d4 3 4 L4 d16
L3 (b) (c) 2
2
2 Y Nbr. Length [m] Reference frame [m]
L2 Body Li di / d̲ i 1 Z 2 0.4432 - / - 1 3 0.4312 - / -
1 4 0.1650 0.0653 / 0.0628 X
(a) Figure 5.6: Lower extremity representation of the biomechanical model with the location of the reference frames of the local coordinates presented in Table 5.1. a) Isometric view of the lower extremity. b) Frontal view of the lower torso. c) Sagittal view of the lower torso.
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The information concerning the physiological and geometric properties of each one of the forty- three muscles modelled are compiled in Table 5.1. The muscle parameters of each muscle are: the maximum isometric force F0 , the pennation angle α0 , the resting length L0 , the tendon length LT , the number of points that describe the muscle path, the number of the bodies to which the muscle is attached (see Chapter 3), and the local coordinates of the origin, via and insertion points.
Table 5.1: Lower extremity muscle apparatus adjusted from Yamaguchi (2001). Described muscle information: action, maximum isometric force, pennation angle, resting and tendon length and attachment points. The illustration of each muscle was extracted from the site ‘Musculoskeletal Atlas’ and it is reproduced in this work with the permission of the authors.
Gluteus Medius: Major abductor of the thigh.
F0 [N] 0 [deg] L0 [m] LT [m] N Pts. Ref. i i [m] i [m] i [m] 4 0.0225 0.0391 0.1195 Anterior 546 8 0.0535 0.078 2 3 -0.0235 0.2010 0.0599 4 -0.0217 0.0531 0.0757 Middle 382 0 0.0845 0.053 2 3 -0.0279 0.2073 0.0569
4 -0.0580 0.0195 0.0640 Posterior 435 19 0.0646 0.053 2 3 -0.0334 0.2085 0.0560
Gluteus Minimus: Abducts and medially rotates the hip joint.
F0 [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 0.0167 0.0012 0.1043 Anterior 180 10 0.068 0.016 2 3 -0.0078 0.2024 0.0605 4 -0.0003 0.0027 0.0979 Middle 190 0 0.056 0.026 2 3 -0.0104 0.2024 0.0605
4 0.0196 0.0029 0.0846 Posterior 215 21 0.038 0.051 2 3 -0.0146 0.2046 0.0594
Gluteus Maximus: Major extensor of the hip joint. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 -0.0553 0.0696 0.0692 Anterior 382 5 0.142 0.125 4 4 -0.0648 0.0103 0.0875 3 -0.0494 0.1868 0.0423
3 -0.0299 0.1525 0.0508 4 -0.0705 0.0265 0.0556 Middle 546 0 0.147 0.127 4 4 -0.0732 -0.0423 0.0903 3 -0.0460 0.1564 0.0316
3 -0.0169 0.1039 0.0453 4 -0.0909 -0.0219 0.0057 Posterior 368 5 0.144 0.145 4 4 -0.0883 -0.0948 0.0398 3 -0.0323 0.1010 0.0146 3 -0.0065 0.0602 0.0444
Adductor Longus: Abducts and flexes the thigh, helps to laterally rotate the hip joint. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 0.0316 0.0735 0.0167 418 6 0.138 0.11 2 3 0.0054 -0.0145 0.0253
55
Table 5.1: (continuation)
Adductor Brevis: Abducts and flexes the thigh, helps to laterally rotate the thigh.
F0 [N] 0 [deg] L0 [m] LT [m] N Pts. Ref. i i [m] i [m] i [m] 4 0.0048 -0.0813 0.0162 286 0 0.133 0.02 2 3 0.0010 0.0844 0.0318
Adductor Magnus: Powerful thigh adductor, superior horizontal fibres also help to flex the thigh, while vertical fibres help extend the thigh. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 -0.0095 -0.1069 0.0252 Superior 346 5 0.087 0.06 2 3 -0.0049 0.0827 0.0366 4 -0.0193 -0.1087 0.0304 Middle 312 3 0.121 0.13 2 3 0.0058 -0.0333 0.0245 4 -0.0134 -0.1076 0.0273 Inferior 444 5 0.131 0.26 2 3 0.0076 -0.2011 -0.0287
Tensor Fasciae Latae: Helps stabilize and steady the hip and knee joints by putting tension on the iliotibial band of fascia. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 0.0321 0.0302 0.1226 155 3 0.095 0.425 4 3 0.0318 0.1061 0.0645 3 0.0058 -0.2240 0.0386 2 0.0062 0.0502 0.0306
Pectineus: Adducts the thigh and flexes the hip joint. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 0.0202 -0.0668 0.0446 177 0 0.133 0.001 2 3 -0.0132 0.1248 0.0273
Iliacus and Psoas: Flex the torso and thigh with respect to each other. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 -0.0038 0.0452 0.0844 Iliacus 429 7 0.1 0.09 5 4 -0.0413 -0.0452 0.0841 4 -0.0337 -0.0709 0.0845 3 0.0018 0.1549 0.0062 3 -0.0208 0.1465 0.0139 4 -0.0011 0.0967 0.0286 Psoas 371 8 0.104 0.13 5 4 0.0393 -0.0472 0.0750 4 0.0339 -0.0704 0.0838 3 -0.0017 0.1588 0.0041 3 -0.0203 0.1491 0.0122
Quadratus Femoris: Rotates the hip laterally, also helps adduct the hip. Gemellus: Rotates the thigh laterally, also helps abduct the flexed thigh. Piriformis: Lateral rotator of the hip joint, also helps abduct the hip if it is flexed. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 -0.0501 -0.1046 0.0514 Quad. Fe. 254 0 0.054 0.024 2 3 -0.0412 0.1748 0.0395 4 -0.0419 -0.0719 0.0705 Gemellus 109 0 0.024 0.039 2 3 -0.0153 0.2100 0.0479 4 -0.0751 0.0094 0.0232 Piriformis 296 10 0.026 0.115 3 4 -0.0551 -0.0182 0.0649 3 -0.0160 0.2097 0.0472
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Table 5.1: (continuation)
Semitendinosus: Extends the thigh and flexes the knee, also rotates the tibia medially, especially when the knee is flexed.
F0 [N] 0 [deg] L0 [m] LT [m] N Pts. Ref. i i [m] i [m] i [m] 4 -0.0594 -0.0940 0.0596 328 5 0.201 0.262 4 2 -0.0324 0.0962 -0.0150 2 -0.0116 0.0755 -0.0253 2 0.0028 0.0539 -0.0199
Semimembranosus: Extends the thigh, flexes the knee, and also rotates the tibia medially, especially when the knee is flexed. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 -0.0550 -0.0912 0.0687 1030 15 0.08 0.359 2 2 -0.0250 0.0971 -0.0200
Biceps Femoris (Long Head): Flexes the knee, and also rotates the tibia laterally, long head also extends the hip joint. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 -0.0601 -0.0898 0.0658 717 0 0.109 0.341 2 2 -0.0083 0.0773 0.0436
Biceps Femoris (Short Head): Flexes the knee, and also rotates the tibia laterally. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 3 0.0054 -0.0145 0.0253 402 23 0.173 0.1 2 2 -0.0104 -0.0777 0.0419
Sartoris: Flexes and laterally rotates the hip joint and flexes the knee. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 0.0477 0.0078 0.1227 104 0 0.579 0.04 5 3 -0.0032 -0.1719 -0.0455 2 -0.0058 0.1092 -0.0411 2 0.0062 0.0917 -0.0395 2 0.0250 0.0658 -0.0260
Gracilis: Flexes the knee, adducts the thigh, and helps to medially rotate the tibia on the femur. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 0.0072 -0.0935 0.0078 108 3 0.352 0.15 3 2 -0.0159 0.1034 -0.0369 2 0.0062 0.0662 -0.0235
Rectus Femoris: Extends the knee and flexes the hip. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 4 0.0337 -0.0216 0.0956 779 5 0.084 0.346 4 2 0.0609 0.1748 0.0035 2 0.0506 0.1313 0.0026 2 0.0403 0.0677 0
57
Table 5.1: (continuation)
Vastus Medialis: Extends the knee.
F0 [N] 0 [deg] L0 [m] LT [m] N Pts. Ref. i i [m] i [m] i [m] 3 0.0151 -0.0132 0.0203 1294 5 0.089 0.126 5 3 0.0385 -0.0856 0.0010 2 0.0549 0.1756 -0.0150 2 0.0506 0.1313 0.0026 2 0.0403 0.0677 0
Vastus Intermedius: Extends the knee. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 3 0.0313 0.0057 0.0355 1365 3 0.087 0.136 5 3 0.0362 -0.0116 0.0308 2 0.0544 0.1792 0.0019 2 0.0506 0.1313 0.0026 2 0.0403 0.0677 0
Vastus Lateralis: Extends the knee. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 3 0.0052 -0.0132 0.0377 1871 5 0.084 0.157 5 3 0.0291 -0.0664 0.0442 3 0.0591 0.1733 0.0170 2 0.0506 0.1313 0.0026 2 0.0403 0.0677 0
Gastrocnemius (Medial and Lateral head): Powerful plantar flexor of the ankle. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 3 -0.0137 -0.2110 -0.0254 Medial 1113 17 0.045 0.408 3 3 -0.0273 -0.2274 -0.0281 2 -0.0224 0.1022 -0.0304 3 -0.0167 -0.2128 0.0294 Lateral 488 8 0.064 0.385 4 3 -0.0303 -0.2311 0.0290 2 -0.0249 0.1028 0.0242
1 -0.0439 0.0242 0.0026
Soleus: Powerful plantar flexor of the ankle. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 -0.0025 -0.0056 0.0073 2839 25 0.03 0.268 2 1 -0.0439 0.0242 0.0026
Tibialis Posterior: Principal invertor of the foot, also adducts the foot, plantar flexes the ankle, helps to supinate the foot. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 -0.0097 0.0134 0.0020 1270 12 0.031 0.31 4 2 -0.0148 -0.2653 -0.0236 1 -0.0070 0.0266 -0.0205 1 0.0281 0.0093 -0.0200
58
Table 5.1: (continuation)
Tibialis Anterior: Dorsiflexor of the ankle and invertor of the foot.
F0 [N] 0 [deg] L0 [m] LT [m] N Pts. Ref. i i [m] i [m] i [m] 2 0.0185 -0.0150 0.0119 603 5 0.098 0.223 3 2 0.0339 -0.2549 -0.0182 1 0.0671 0.0111 -0.0224
Flexor Digitorum Longus: Flexes the toes 2-5, also helps in plantar flexion of the ankle. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 -0.0086 -0.0585 -0.0019 310 7 0.034 0.4 5 2 -0.0159 -0.2653 -0.0202 1 -0.0051 0.0246 -0.0199 1 0.0218 0.0110 -0.0182 1 0.1157 -0.0145 0.0193
Flexor Hallucis Longus: Flexes the great toe, helps to supinate the ankle, very weak plantar flexor of the ankle. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 -0.0081 -0.0882 0.0252 322 10 0.043 0.38 5 2 -0.0192 -0.2681 -0.0179 1 -0.0113 0.0209 -0.0160 1 0.0544 0.0348 -0.0175
1 0.1224 -0.0188 -0.0188
Extensor Digitorum Longus: Extends the toes 2-5 and dorsiflexes the ankle. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 0.0033 0.1000 0.0284 341 8 0.102 0.345 4 2 0.0298 -0.2606 0.0074 1 0.0429 0.0318 0.0077 1 0.1116 -0.0010 0.0207
Extensor Hallucis Longus: Extends the great toe and dorsiflexes the ankle [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 0.0012 -0.0297 0.0235 108 6 0.111 0.305 5 2 0.0336 -0.2584 -0.0088 1 0.0477 0.0319 -0.0131 1 0.0796 0.0240 -0.0176 1 0.1232 0.0073 -0.0199
Peroneus Brevis: Everts the foot and plantar flexes the ankle. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 -0.0072 -0.1203 0.0355 348 5 0.05 0.161 5 2 -0.0204 -0.2789 0.0292 2 -0.0148 -0.2903 0.0298 1 -0.0017 0.0203 0.0309 1 0.0187 0.0152 0.0417
Peroneus Tertius: Works with the extensor digitorum longus to dorsiflex, evert and abduct the foot. [N] [deg] [m] [m] N Pts. Ref. i [m] [m] [m] 2 0.0010 -0.1366 0.0238 90 13 0.079 0.1 3 2 0.0236 -0.2670 0.0164 1 0.0365 0.0160 0.0374
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Table 5.1: (continuation) Peroneus Longus: Everts the foot and plantar flexes the ankle; also helps to support the transverse arch of the foot.
F0 [N] 0 [deg] L0 [m] LT [m] N Pts. Ref. i i [m] i [m] i [m] 2 0.0005 -0.0092 0.0373 754 10 0.049 0.345 7 2 -0.0213 -0.2810 0.0295 2 -0.0167 -0.2928 0.0298 1 -0.0049 0.0163 0.0297 1 0.0191 0.0040 0.0359
1 0.0360 0.0004 0.0195 1 0.0707 0.0019 -0.0104
The Yamaguchi’s biomechanical model is set in a standing position with the muscles applied in the right leg in order to determine the following variables: muscle-tendon length Lmt , current pennation angle α and muscle length Lm. The muscle-tendon length is given by the expression:
mtm LLLcos() T (5.13)
The pennation angle α can be determined for different dimensions of the muscle-tendon using the expression (Oliveira et al., 2015):
L00sin() arctanmt (5.14) LL T where L0 and α0 correspond, respectively to the length of the muscle and pennation angle when the muscle is resting and LT is the length of the tendon, which are described in Table 5.1. Subsequently, the current length of the muscle can be easily obtained by the expression:
mt m LL L T (5.15) cos( )
m The objective is to collect the pennation angle α and the two dimensionless ratios LT/L0 and L /L0. These ratios are assumed to be constant (Quental et al., 2015) for the standing position of the Yamaguchi’s model and, under this assumption, the new musclulotendon parameters associated to the subject are calculated. The biomechanical model is scaled to the subject dimensions, applying the procedure described in Section 3.4, and the resultant dimension scaling factor is also used to scaled the local coordinates of each muscle associated to the reference anatomical segment. The individual total
mt muscle length of the subject Lsubject is calculated and the Equations (5.14) and (5.15) are used to calculate
subject subject the subject’s muscles resting L0 and tendon LT lengths:
Lmt L LLLsubjectsubject and subjectT subject (5.16) 00Lm L T L cos( ) T 0 LL00
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At this point the subject’s muscles shortening velocity L̇ m(t) can be calculated. The muscle-tendon shortening velocity L̇ mt(t) is obtained through cubic spline interpolation and differentiation of the total muscle length Lmt(t), and is used to determine the muscle shortening velocity L̇ m(t). The time differentiation of Equation (5.13) leads to:
Ltmt () Ltm () (5.17) cos()sin()tan()
The complete lower muscle apparatus can be inserted into the right leg of the biomechanical model, which is illustrated in Figure 5.7.
Figure 5.7: Biomechanical model with the lower extremity muscle apparatus applied in the right leg. Visualization generated in MATLAB.
5.4 Optimization
The introduction of muscle actuators into the biomechanical system leads to an indeterminate inverse dynamic problem since there are more unknowns to determine than equations of motion. From the infinite set of possible solutions, only one describes the actual muscle activation pattern of the subject under analysis while respecting the constraint equations. In this work, an optimization procedure is implemented, to obtain the muscle forces developed by the lower extremity muscle apparatus. The optimization procedure finds the solution that minimizes a prescribed physiological criterion while ensuring the fulfilment of the equations of motion and the physiological properties of the muscles. From an instant of time t, it is mathematically formulated as (Arora, 2007):
61
Given: x xi
Minimize:() F0 x (5.18) f ()0x Subjected to: lowerupper xxxiii
where x is a vector containing the design variables, F0 ()x is the objective function to be minimized,
lowerupper f ()x are the constraint equations that the control variables must satisfy and xiii x x are the boundaries of the control variables. In the context of the biomechanical problem presented here represents the equations of motion, and are zero and one, respectively, for the control variables related to the muscle activations. From the several objective functions (Silva, 2003; Tsirakos et al., 1997) capable of simulating the central nervous system decisions, the two cost-functions applied here are among the most commonly used in human locomotion applications (Collins, 1995; Crownshield and Brand. 1981):
(1) Sum of the Square of the Individual Muscle Forces
2 nnmm mama 2 FF mLm L (5.19) FFa0 ()x CE m mm11F0
(2) Sum of the Cube of the Individual Average Muscle Stresses
3 nnmm ma2 ma FF m LL m (5.20) Fa0 ()x CE m2 mm11F0
where nma is the number of muscle actuators and is the specific muscle strength with a constant value of 31.39 N/cm2 (Yamaguchi, 2001). During the inverse dynamic analysis, the equations of motion must always be satisfied. The passive component of the muscle forces is introduced into the vector of forces g using Equation (5.11). Accordingly, for an instant of time t, the vector holding the equality constraints is defined as:
f 1 f Φq λ () Mq g 0 (5.21) fnc
The gradients of Equation (5.21) with respect to the control variables are easily determined, leading to:
f f Φ (5.22) λqλ where Φq is the Jacobian matrix, already calculated for Equation (5.21). The resultant non-linear problem is solved using the interior-point algorithm of the fmincon function of MATLAB.
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5.5 Application Case to a Normal Gait Cycle
The muscle activations and muscle forces presented in this Section are computed for the gait motion described in Chapter 4, which is recorded at a sampling frequency of 100Hz and lasted 1.11s. The optimization procedure is applied independently for each instant of time, i.e., using a static optimization approach. The muscle activations calculated using, alternatively, the objective functions described by Equations (5.19) and (5.20) are presented in Figure 5.8. The optimization algorithm fails to find a feasible solution for 15 instants of time, between 42% and 57% of the gait cycle. It was found that the constraint equations related to the kinematic joint of the right foot could not be satisfied, regardless of the muscle forces produced. Furthermore, unusual muscle activations are observed during this gait period. In particular, some muscles show short periods of full activation that are inconsistent with the performance of a normal gait cycle.
Gluteus Minimus Middle 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride
Rectus Femoris 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Figure 5.8: Muscle activation patterns using two cost-functions: sum of the individual muscle forces (SINMF) (Collins, 1995) and sum of the cube of the individual muscle stresses (CIAMS) (Crowninshield and Brand, 1981). Feasible solutions were not found for the range of frames between 42% and 57% of the gait cycle.
In the work of Rajagopal et al. (2016), the maximum isometric force of each muscle is increased to allow the biomechanical model to produce muscle-generated joint moments large enough to match those produced in the dynamic motion. Considering that there is a large variability in the muscle properties presented in the literature (Carbone et al., 2012), a similar approach is followed in this work to check if larger muscle forces can overcome the violation of the equations of motion and solve the complete gait motion cycle. A two-fold increase in the maximum isometric forces is enough to allow
63 the biomechanical model to solve all gait cycle. The muscle activations obtained for all the muscles composing the lower extremity muscle apparatus are presented in Figure 5.9.
Gluteus Medius Anterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gluteus Medius Middle 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gluteus Medius Posterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gluteus Minimus Anterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gluteus Minimus Middle 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gluteus Minimus Posterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gluteus Maximus Anterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Figure 5.9: (continuation)
64
Gluteus Maximus Middle 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gluteus Maximus Posterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Adductur Longus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Adductur Brevis 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Adductur Magnus Superior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Adductur Magnus Middle 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Adductur Magnus Inferior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Tensor Fasciae Latae 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Figure 5.9: (continuation)
65
Pectineus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Iliacus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Psoas 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Quadratus Femoris 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gemellus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Piriformis 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Semitendinosus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Semimembranosus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Figure 5.9: (continuation)
66
Biceps Femoris Long 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Biceps Femoris Short 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Sartoris 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gracilis 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Rectus Femoris 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Vastus Medialis 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Vastus Intermedius 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Vastus Lateralis 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Figure 5.9: (continuation)
67
Gastrocnemius Medial 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Gastrocnemius Lateral 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Soleus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Tibialis Posterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Tibialis Anterior 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Flexor Digitorum Longus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Flexor Hallicus Longus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Extensor Digitorum Longus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Figure 5.9: (continuation)
68
Extensor Hallicus Longus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Peroneus Brevis 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Peroneus Longus 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Peroneus Tertius 1 SINMF
0.5 CIAMS a(t)
0 0 20 40 60 80 100 % of Stride Figure 5.9: Activation patterns for the muscles of the locomotor apparatus. Comparison of two cost- functions: sum of the individual muscle forces (SINMF) (Collins, 1995) and sum of the cube of the individual muscle stresses (CIAMS) (Crowninshield and Brand, 1981).
The fact that a two-fold increase in the maximum isometric forces ensures always a feasible solution suggests that the muscle properties considered may not reflect the actual strength of the lower extremity muscle apparatus or the geometric description of the muscles may be underestimating the muscle moment arms. Note that even with such increase, there are still some muscles that show continuous maximum activation, such as the Sartoris or the Gastrocnemius Medial muscles, which is not expected to occur for a normal gait cycle (Simpson et al., 2015). The muscle activations obtained using the two cost-functions show large differences in some muscles, being the most evident: Tensor Fasciae Latae, Gemellus, Sartoris, Gracilis, Gastrocnemius Medial and Lateral and Flexor Hallicus Longus muscles. While other muscle shows predominant null muscle activations considering both cost-functions, namely: Adductor Brevis, Adductor Magnus Superior, Middle and Inferior, Quadratus Femoris, Vastus Medialis, Intermedius and Lateralis, Tibialis Anterior, Extensor Digitorum Longus, Peroneus Tertius. These results do not compare well with those reported in the recent literature (Rajagopal, 2016; Ambrósio and Siva, 2005; Simpson et al., 2015).
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Although most of the results presented in the literature do not show all muscle activations and some use different activation boundaries (Anderson and Pandy, 2001a), in the work of Ambrósio and Siva (2005), all the muscle activations and muscle forces are presented and the muscle activations are bounded between zero and one. In this sense the work reported by Ambrósio and Siva (2005) also show full saturation of some of the muscle activations, in particular of the Tensor Fasciae Latae, Gastrocnemius Medial and Lateral, Biceps Femoris among others, which is similar to the behavior observed here. Concerning the muscle forces, these correspond to the sum of the two force components: active and passive. Knowing the muscle activations, the active component of the muscle forces is calculated using Equation (5.8). The passive muscle forces, which were previously calculated before the optimization procedure, are determined using Equation (5.11). The muscle forces obtained for all the muscles composing the lower extremity muscle apparatus are presented in Figures 5.10.
Gluteus Medius Anterior 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gluteus Medius Middle 400 SINMF
200 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gluteus Medius Posterior 400 SINMF
200 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gluteus Minimus Anterior 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Figure 5.10: (continuation)
70
Gluteus Minimus Middle 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gluteus Minimus Posterior 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gluteus Maximus Anterior 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gluteus Maximus Middle 100 SINMF
50 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gluteus Maximus Posterior 100 SINMF
50 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Adductur Longus 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Adductur Brevis 40 SINMF
20 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Adductur Magnus Superior 40 SINMF
20 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Figure 5.10: (continuation)
71
Adductur Magnus Middle 40 SINMF
20 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Adductur Magnus Inferior 40 SINMF
20 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Tensor Fasciae Latae 400 SINMF
200 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Pectineus 100 SINMF
50 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Iliacus 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Psoas 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Quadratus Femoris 100 SINMF
50 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gemellus 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Figure 5.10: (continuation)
72
Piriformis 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Semitendinosus 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Semimembranosus 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Biceps Femoris Long 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Biceps Femoris Short 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Sartoris 400 SINMF
200 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gracilis 400 SINMF
200 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Rectus Femoris 2000 SINMF
1000 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Figure 5.10: (continuation)
73
Vastus Medialis 500 SINMF
CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Vastus Intermedius 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Vastus Lateralis 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gastrocnemius Medial 4000 SINMF
2000 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Gastrocnemius Lateral 2000 SINMF
1000 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Soleus 4000 SINMF
2000 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Tibialis Posterior 1000 SINMF
500 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Tibialis Anterior 100 SINMF
50 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Figure 5.10: (continuation)
74
Flexor Digitorum Longus 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Flexor Hallicus Longus 400 SINMF
200 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Extensor Digitorum Longus 40 SINMF
20 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Extensor Hallicus Longus 40 SINMF
20 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Peroneus Brevis 200 SINMF
100 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Peroneus Longus 400 SINMF
200 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Peroneus Tertius 40 SINMF
20 CIAMS Fm [N] Fm 0 0 20 40 60 80 100 % of Stride Figure 5.10: Muscle forces for the muscles of the locomotor apparatus. Comparison of two cost- functions: sum of the individual muscle forces (SINMF) (Collins, 1995) and sum of the cube of the individual muscle stresses (CIAMS) (Crowninshield and Brand, 1981).
As a consequence of the muscle activations obtained by the two cost-functions, the resultant muscle forces show no agreement with the literature (Ambrósio and Siva, 2005).
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5.6 Discussion
The muscle activations and muscle forces for the lower extremity apparatus are estimated using two different cost-functions. Some muscles display almost no activation, while other muscles, show full activation. Compared to the literature significant differences are obtained that are not expected for a normal gait cycle. There are two possible main sources for the discrepancies that may be influencing the results: either the representation of the muscle moment arms or the scaling of the muscle parameters.
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6 Conclusion and Future Development
6.1 Conclusions
The human locomotion apparatus is a highly complex system, which is studied in this work from the biomechanical point of view by using multibody based methodologies. A three-dimensional biomechanical model of the whole human body was formulated using Cartesian coordinates with the objective of studying the mechanical behaviour of a normal gait cycle. The data and structure of the biomechanical model are based in the work of Silva and Ambrósio (2003). The biomechanical model consists in a full body representation of the human based on the 50th percentile human male geometrical and physical properties. The anatomical articulations are approximated by ideal mechanic joints and their relative motion is acted by driver actuators. The scaling procedure used to fit the biomechanical model to the dimensional and physical characteristics of the subject under analysis is also described here. To perform an inverse dynamic analysis of a normal gait cycle, a motion acquisition, including anthropometric, kinematic and kinetic data, was performed first. These data were properly treated using a filtering technique to remove the noise related to the position of the anatomical markers, as well as of the components and centre of pressure of the ground reaction forces. A kinematic consistency analysis was also performed to ensure the calculation of consistent positions. Two inverse dynamic problems were solved considering the biomechanical model actuated only by driver actuators, referred to as determinate problem, and considering the model actuated by muscles as well, referred to as indeterminate problem. The results of the determinate problem, namely, the joint torques, showed some differences from the literature, which requires further scrutiny to identify the source of discrepancies that could be affecting the inverse dynamic solution. In the process of solving the determinate inverse dynamic problem, particular attention was paid to the formulation of drivers for the spherical joints. A novel formulation for a spherical joint, corresponding to the use of three equivalent torque drivers, is proposed when using Cartesian coordinates. For the indeterminate problem, the driver actuators of the right leg were removed, specifically, the drivers at the hip, knee and ankle, and replaced by muscle actuators while keeping the remaining drivers to control the biomechanical model. The muscle actuators behaviour was modelled by a Hill- type muscle model, neglecting the activation dynamics and considering the tendon to be rigid. The indeterminate problem was solved by a static optimization procedure to predict the set of muscle activations responsible for the observed gait. The results of the indeterminate problem, the muscle activations predicted by the two objective functions used and the resultant muscle forces do not compare well with the literature. For some instants of time the optimization algorithm could not find a feasible solution. Considering a two-fold, increase
77 in the maximum isometric force, all gait cycle was solved, but the main source of numerical problems behind the implementation of muscles in the right leg was not identified. More analysis cases must be performed to support the finding for the problems found in the solution of inverse dynamic problem, in order to identify if the source of any potential problem is the muscle database used, the biomechanical model developed, the solution method implemented or the subject data acquired.
6.2 Future Developments
When constructing a biomechanical model, defining the kinematic joints and drivers, implementing the lower extremity muscles and calculating the solution of both determinate and indeterminate inverse dynamic problems, several approximations and assumptions are considered to simplify and make the computation of the results possible. Concerning the results of the inverse dynamic analysis of the determinate and indeterminate biomechanical systems, several tests should be performed to find the sources for the discrepancies in these results. For the case of the determinate solution, the methods to process the kinematic input data should be revised, specifically, the kinematic consistent analysis. For instance, the hip joint centre should be determined by a functional method and compared with the predictive method applied in this work to check for differences in the joint torques calculated. For the case of the indeterminate solution, the implementation of the lower extremity muscles in the right leg should also be verified. An analysis of the muscle moment arms should be performed to assess the correct insertion of the muscles across the anatomical segments as well as the appropriate scaling of the muscle properties. Additionally, the methodologies developed can be applied to a larger set of motions and subjects in order to obtain a representative set of results, with the objective of developing a collection of results for statistical study. On the gait analyses should only be obtained results on the analyses of a wider set of acquired motions to decrease the uncertainties during the motion acquisition. The biomechanical model presented can be adapted to study other types of gait analysis, such as pathological gait or the gait of subjects with prosthesis or orthoses, to show the differences in the mechanical response. Nevertheless, the detail of the biomechanical model can also be increased, i.e., for instance the foot can be subdivided in order to represent the dorsi-flexion of the toes, the muscle system ca be modelled by using obstacle wrapping approaches instead of via points and, eventually, discretizing some muscles in several segments. The muscle activations calculated by the optimization procedure used in this work can be compared with EMG data collected at the LBL to assess which one of the two physiological criteria better predicts the muscle activations. Regarding the muscle model presented, more realistic and accurate simulations may be achieved by including in the model the activation dynamics and a compliant tendon model (Millard, 2013;
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Oliveira et al., 2015). The activation dynamics introduces the intricate interaction involving the central nervous system and the musculoskeletal systems. Conversely, the compliant tendon embeds the interaction between the muscular and skeletal system. Certainly, the solution of the biomechanical model with the complete activation and contraction dynamics imply the use of a global optimization (Rodrigo et al., 2008), the Extended Inverse Dynamics (Ackermann, 2007) or the Window Moving Inverse Dynamics Optimization (Quental, 2016) for the solution of the indeterminate inverse dynamic analysis. The use of these solution methodologies also allows for the use of time dependent physiological merit functions such as the metabolic energy (Ackermann, 2006) to represent the objective function. The muscle database can be broader with distinguished sets of lower extremity muscles, each one describing the respective musculotendon parameters of different musculoskeletal models (Carbone et al., 2015; Rajagopal et al., 2016). This can shorten the gap between the subject under analysis and the biomechanical model by applying the set of lower muscles that better described the subject. The introduction of muscles into the upper limbs of the model is another future development that should be considered to investigate the muscle forces developed in the upper part of the body during a gait cycle. The swing of the arms and the tilt between the lower and upper torso is an important aspect of the gait and the muscle forces there produced can also be studied.
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Appendix A – Marker Set Protocol
The anatomical markers used to perform a kinematic data acquisition of human gait in the Lisbon Biomechanics Laboratory are described in Table A1 and illustrated in Figure A1. The protocol considered follows the ISB recommendations (Wu et al., 2002, 2005).
Nbr. Point Description 1 RF_Meta1 Right foot metatarsal 1 2 RF_Meta5 Right foot metatarsal 5 3 RF_Calcaneus Right foot calcaneus 4 RF_MM Right foot – Right lower leg medial malleolus 5 RF_LM Right foot – Right lower leg lateral malleolus 6 RLL_MC_FE Right lower leg medial condyle – Right upper leg femoral epicondyle 7 RLL_LC_FE Right lower leg lateral condyle – Right upper leg femoral epicondyle 8 RUL_Tro Right upper leg trochanter 9 LT_RASIS Lower torso right ASIS 10 LT_LASIS Lower torso left ASIS 11 LT_RPSIS Lower torso right PSIS 12 LT_LPSIS Lower torso left PSIS 13 LUL_Tro Left upper leg trochanter 14 LLL_MC_FE Left lower leg medial condyle – Left upper leg femoral epicondyle 15 LLL_LC_FE Left lower leg lateral condyle – Left upper leg femoral epicondyle 16 LF_Meta1 Left foot metatarsal 1 17 LF_Meta5 Left foot metatarsal 5 18 LF_Calcaneus Left foot calcaneus 19 LF_MM Left foot – Left lower leg medial malleolus 20 LF_LM Left foot – Left lower leg lateral malleolus 21 RH_Meta5 Right hand metacarpal 5 22 RH_Meta2 Right hand metacarpal 2 23 RH_US Right hand ulnar styloid 24 RH_RS Right hand radial styloid 25 RLA_EM Right lower arm medial epicondyle 26 RLA_EL Right lower arm lateral epicondyle 27 RUA_AC Right upper arm acromioclavicular joint 28 UT_IJ Upper torso Incisura jugulars 29 UT_C7 Upper torso 7th cervical vertebra 30 UT_Belly_button Upper torso above belly button 31 UT_ T12 Upper torso 12th thoracic vertebra or 1st lombar vertebra 32 LUA_AC Left upper arm acromioclavicular joint 33 LLA_EM Left lower arm medial epicondyle 34 LLA_EL Left lower arm lateral epicondyle 35 LH_Meta5 Left hand metacarpal 5 36 LH_Meta2 Left hand metacarpal 2 37 LH_US Left hand ulnar styloid 38 LH_RS Left hand radial styloid 39 Head_RForehead Head right forehead 40 Head_LForehead Head left forehead 41 Head_Chin Head chin 42 Head_Front Head frontal 43 Head_ROccipital Head right occipital 44 Head_LOccipital Head left occipital
Table A1: Markers description and location.
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H_Front
H_RFH H_LFH H_LOcc H_ROcc
UT_IJ RUA_AC LUA_AC UT_C7
H_Chin
UT_BB
RLA_EM LLA_EM UT_T12 RLA_EL LLA_EL
LT_RASIS LT_LASIS RH_US RUL_Tro LUL_Tro RH_RS
RH_Meta2
LT_LPSIS LT_RPSIS RH_Meta5 RLL_MC_FE LLL_MC_FE RLL_LC_FE LLL_LC_FE
RF_MM LF_MM RF_LM LF_LM LF_Calcaneus RF_Calcaneus RF_Meta5 LF_Meta5
RF_Meta1 LF_Meta1 Figure A1: Frontal, sagittal and anterior view of the human skeleton. Location of the forty-four markers placed on the subject’s skin.
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Appendix B – Visualization of the Biomechanical Model Using SAGA
In the course of the thesis three SAGA models are developed to make a visual assessment of the solution of the kinematic analysis and to identify any obvious error in the implementation of the biomechanical model. However, only one male subject is analysed in this work, another gait analysis was performed with a female individual, and therefore, two models for each gender were created.
(a) (b) Figure B1: One gait cycle illustrated by the female model of SAGA a) Sagittal view. right heel contact at 0%, right toe-off at 60% and again right heel contact at 100%; b) frontal view.
(a) (b) Figure B2: One gait cycle illustrated by the male model of SAGA a) Sagittal view. right heel contact at 0%, right toe-off at 60% and again right heel contact at 100%; b) frontal view.
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(a) (b) Figure B3: One gait cycle illustrated by the skeleton model of SAGA a) Sagittal view. right heel contact at 0%, right toe-off at 60% and again right heel contact at 100%; b) frontal view.
(a) (b) Figure B4: One gait cycle illustrated by the skeleton with lower extremity muscles model of SAGA a) Sagittal view. right heel contact at 0%, right toe-off at 60% and again right heel contact at 100%; b) frontal view.
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