Dynamic Modeling and Analysis of Human Locomotion .Dynamic Modeling and Analysis of Human Locomotion

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  • International Journal for Computational Vision and Biomechanics, Vol. 2, No. 2, July-December 2009

    Serials Publications 2009 ISSN 0973-6778

    Dynamic Modeling and Analysis of Human Locomotion usingMultibody System Methodologies

    Fernando Meireles1, Margarida Machado1, Miguel Silva2 Paulo Flores11Departamento de Engenharia Mecnica, Universidade do Minho, Campus de Azurm, 4800-058 Guimares, Portugal2Departamento de Engenharia Mecnica, Instituto Superior Tcnico, IST/IDMEC, Av. Rovisco Pais1, 1049-001 Lisboa, Portugal

    This paper deals with the development and implementation of a general and comprehensive methodology for dynamicmodeling and analysis of human locomotion based on the multibody systems approach. It is known that the dynamic analysisof the human gait usually requires three types of input data namely: (i) anthropomorphic information regarding the dimensionsand weight of the anatomical segments; (ii) kinematic information used to describe its motion in a unique manner; (iii) andkinetic information describing all external applied forces to the biomechanical model and their respective points of application.In this work the kinematic equations of motion and the full characterization of the biomodels are developed by using Cartesiancoordinates. In particular, the biomechanical model used here composed by seven rigid bodies, corresponding to the mainanatomical segments. The bodies are connected by six revolute joints. The trajectories of all the bodies that represent thesystems degrees of freedom have guide constraints associated, which are obtained experimentally. The motion data of thebiomechanical system consists of the trajectory of a set of anatomical points located at the natural joints. The collected datapoints are then interpolated using cubic splines in order to define the necessary mathematical expressions that representthat guide constraint equations. Furthermore, a set of external forces applied to the biomodel as well as respective points ofapplication are acquired by a pressure platform that reads the reaction forces between the ground and feet segments. Themethodologies developed here are implemented into computational codes devoted to the kinematic and dynamic analysis ofgeneral multibody systems. Finally, some results are presented and analyzed in order to discuss the main procedures andassumptions adopted in this work.

    Keywords: Multibody systems, Dynamics, Biomechanical model, Human locomotion

    1. INTRODUCTIONIn a broad sense, the biomechanics of motion has threefundamental objectives. The first one is related to themotion variables identification of the system underanalysis. The second objective seeks to understand thedependency between the motion variables and thebiomodel performance. Finally, biomechanics aims toapply design and optimization procedures ofbiomechanical systems. Thus, taking into account thedevelopment of science and engineering towards peopleswelfare, it becomes essential to develop computationaltools that are able to predict the kinematic and dynamicbehavior of biomechanical systems [1, 2]. It is well knownthat computational simulation of the human motionrequires that development and implementation of suitablemathematical models that correctly describe the behaviorof the human body and its interaction with thesurrounding environment. In a general way, there areessentially two major classes of biomechanical modelsnamely: (i) the models described using finite elementmethods and (ii) the models developed under theframework of multibody systems formulations. Finiteelement methods are applied in cases where localizedstructural deformations of soft tissues need to be analyzedin detail, while the multibody models are usually used in

    cases where gross-motions are involved and whencomplex interactions with surrounding environment areexpected. Human gait as a gross-motion simulation is, ingeneral, described using multibody systemsmethodologies [1].

    Over the last years, the study and characterization ofthe human body motion has deserved the attention of theacademic and scientific communities [3-5]. In thesestudies, the geometrical and physical properties of bones,muscles, tendons, etc., that constitute the biomechanicalmodels have been taken into account in the developmentof computational approaches. In addition, the correlationbetween numerical and experimental results has beenstudied by a good number of research works [6-8]. Thestudy of human body motion as a multibody systems is achallenging research field that has undergone enormousrecent developments that finds application in severalactivities, such as (i) analysis of top athletic actions toimprove different sporting performances; (ii) optimizationof the design of sportive equipment; (iii) ergonomicstudies to assess operating conditions for complex andefficiency in different aspects of the human bodyinteractions with the environment; (iv) orthopedics toimprove the design and analysis of medical auxiliarydevices; (v) occupant dynamic analysis for

    International Journal of Computational Vision and Biomechanics Vol.2 No. 2 (July-December, 2016)

  • 200 International Journal for Computational Vision and Biomechanics

    crashworthiness and vehicle safety related research anddesign; (vi) gait analysis for generation of normal gaitpatterns and consequent diagnosis of pathologies anddisabilities [9-14].

    The present work deals with the development andimplementation of a general and comprehensivemethodology to dynamic modeling and analysis ofbiomechanical systems using a two-dimensionalmultibody systems approach. Although, the six majordeterminants of human gait (hip, knee and ankle flexion-extension, pelvic rotation, pelvic list and lateral pelvicdisplacement) can only be studied in three-dimensionalmodels, the development of two-dimensional models ismuch more simple and can still be useful concerning tothe evaluation of the joint flexion-extension motions andjoint sagittal moments. This is true since the goal of thelocomotion is to propel the body in the plane ofprogression, while supporting the body against the gravityaction, the major motion portion of analysis performedin the sagittal plane. In this work, the equations of motionand the human biomodel description are developed byusing Cartesian coordinates. The trajectories of theanatomical segments that guide the biomodel are obtainedfrom experimental data acquisition, in which relevantanatomical points of reference are marked and followedduring the systems motion. These points are typicallyrepresentative of natural joints. After obtaining the datarelative to the human gait motion, the set of pointsacquired are filtered and then interpolated using cubicsplines functions in order to define the necessarymathematical expressions that represent the guideconstraint equations. Numerical results obtained fromsome computational simulations are used to discuss theassumptions and procedures adopted in this study.


    In order to analyze the dynamic response of a constrainedmultibody system, it is first necessary to formulate theequations of motion that govern its behavior. Theformulation of multibody system dynamics adopted inthis work follows closely that of Nikravesh [15], in whichthe generalized Cartesian coordinates and the Newton-Eulers approach are employed to derive the systemsequations of motion. Figure 1 depicts a multibody system,which consists of a collection of rigid and/or flexiblebodies interconnected by kinematic joints and possiblysome force elements. The kinematic joints control therelative motion between the bodies, while the forceelements represent the internal forces that developbetween bodies due to their relative motion. The forcesapplied over the system components may be the result ofsprings, dampers, actuators or external forces, such asthose developed between colliding bodies.

    When the configuration of a multibody system isdescribed by nc Cartesian coordinates, then a set of malgebraic kinematic independent constraints can bewritten in a compact form as [15],

    (q, t) = 0 (1)where q is the vector of generalized coordinates and t isthe time variable.

    The velocities and accelerations of the systemelements are evaluated using the velocity and accelerationconstraint equations. Thus, the first time derivative withrespect to time of Equation (1) provides the velocityconstraint equations,

    t q q (2)

    where q is the Jacobian matrix of the constraint

    equations, q is the vector of generalized velocities and is the right hand side of velocity equations.

    A second differentiation of Equation (1) with respectto time leads to the acceleration constraint equations,obtained as,

    t tt( ) 2 q q q q q q q q (3)

    where q is the acceleration vector and is the right handside of acceleration equations.

    The equations of motion for a constrained multibodysystem of rigid bodies are written as [15],

    ( ) cMq g g (4)

    where M is the system mass matrix, q is the vector thatcontains the system accelerations, g is the generalizedforce vector and g(c) is the vector of constraint reactionequations. The joint reaction forces can be expressed interms of the Jacobian matrix of the constraint equationsand the vector of Lagrange multipliers as [15],

    ( ) T c qg (5)

    Figure 1: Generic configuration of a multibody system

    Revolute joint

    Spherical joint


    Body 1

    Body n

    Body 3

    Ground body

    Revolute jointwith clearance


    Lubricated joint

    Contact bodies

    Applied torque

    Flexible bodyGravitational forcesOther applied forces

    Body i

    Body 2

    Revolute joint

    Spherical joint


    Body 1

    Body n

    Body 3

    Ground body

    Revolute jointwith cleara