Available online at www.sciencedirect.com
Procedia IUTAM 2 (2011) 86–92
2011 Symposium on Human Body Dynamics
Integrating efficient kinematics in biomechanics of human motions Andres Kecskemethya,∗ a University
of Duisburg-Essen, Chair for Mechanics and Robotics, Lotharstr. 1, 47058 Duisburg, Germany
Abstract Biomechanical modeling of human motion often is based on simple tree-type structures with elementary joint and contact situations. This is adequate for coarse evaluation of motion parameters and their effects but insufficient for detailed analysis of joint or system/environment interactions. In this lecture, an object-oriented approach for the modeling of kinematics and dynamics of musculoskeletal motion is presented which is open for integration of arbitrarily complex subsystems and their coupling to state-of-the art numerical and visualization tools. The approach is based on the concept of kinetostatic transmission element which allows one to embed intricate kinematical dependencies in easy-to-use, multilayered objects. Based on this approach, the inverse and direct dynamics problem can be formulated in a highly-efficient and implementation-independent manner, featuring different methodologies as ’flavors’ of the generic transmission properties. This allows one to re-use methods developed for technical systems such as contact effects, multiloop transmission mechanisms, multidisciplinary mechatronic components, and efficient numerical solution schemes. The concepts have been implemented in the object-oriented 3D biomechanical simulation environment MobileBody featuring mechanism surrogates for joint motion, higher-order foot-ground contact, data fusion of MRI and motion capturing, composite numerical clinical scoring algorithms, optimization-based muscle activation identification, verification by interval analysis, and embedded 3D visualization. c 2011 Published by Elsevier Ltd. Peer-review under responsibility of John McPhee and J´ozsef K¨ovecses Keywords: Kinematics; gait analysis; biomechanics
1. Introduction Biomechanical analysis of human gait motion with stereophotogrammetry is a well-established technique [1, 2, 3]. For this method, small reflective markers are attached to the subject, lighted, and monitored by at least two motion cameras. From the two-dimensional camera images, the position of the markers in space is reconstructed, leading to marker trajectories ri (t) which are stored for later bone motion reconstruction. In gait analysis, the contact wrenches between feet and ground are measured simultaneously with force plates, from which the center of pressure and the ∗ Corresponding
author. Tel: +49-203-379-3344; fax: +49-203-379-2494 Email address:
[email protected] (Andres Kecskemethy)
2210-9838 © 2011 Published by Elsevier Ltd. doi:10.1016/j.piutam.2011.04.009
Andres Kecskemethy / Procedia IUTAM 2 (2011) 86–92
magnitude of the ground reaction force are computed. Techniques for determining marker positions and ground forces are mature and well-established and sagittal plane motion can be estimated from this data with good repeatability. Hence current gait analysis methods are well suited for quantified evaluation of the therapeutic outcome, if a relative improvement needs to be verified objectively [4]. For estimating parameters that can not be measured directly and for reliable prognosis and surgery planning, multibody models of the musculoskeletal system are required [5, 6]. Current implementations of these models were done using commercial, general purpose multibody simulation systems like ADAMS [7, 8] or by specialized programs such as SIMM ([9]) or AnyBody ([10]). These models can not predict kinematics and statics of the muscles properly, unless accurate estimates of the relative joint motion are available, which is still an unsolved problem. As an example, the identified dynamic bone motion is influenced significantly by the marker protocol [11, 12, 13], marker placement and skin artifacts [14, 15, 16]. In addition, unknown dimensions and inertia properties of the musculoskeletal system are often computed by scaling anthropometric data from single specimen measurements [6, 17]. This leads to significant errors, as shown by [18] for the case of inertia parameters of children’s leg segments. These errors are avoided by using patient-specific computer tomography data [19], from which bone geometry can be segmented easily, but exposes the subject to ionizing radiation. Another open question is the identification of muscle force recruitment ([20], for which a full forward dynamics model of the human motion and detailed cost functions (see, e.g. [21]) are required, and for which again fast computer code is necessary. 2. Materials and methods The present approach employs an object-oriented implementation of the multibody dynamics of human motion called ’MobileBody’ to realize an embeddable, highly efficient representation of forwards or inverse dynamics (fig. 1). a a The modeling is based on the C++ library M a aBILE [22] which has been already employed for modeling, simulation and design of technical systems, including robotics ([23]), mining machinery ([24]), up to roller coasters ([25]). In the setting of human motion reconstruction, the package could be used to integrate geometrical data from magnet resonance imaging (MRI) in conjunction with a semi-automatic model-based segmentation algorithm into the model ([26]), to derive efficient multibody code for muscle recruitment analysis using forward dynamic and considering verification issues ([27]), and also to model the foot/ground contact using simplified flat cylinder-plane contact geometries [28]. Recently, the library has also been extended to perform computations with intervals, thus allowing to track errors using methods of interval analysis ([29]). Such verified computations may be useful to detect the areas in which results can lie. By the use of object-oriented notions, it is easy to define embeddable models for the dynamics which can be extended to include new types of features and also to be coupled with other disciplines. In contrast to the closed simulation systems, this offers new possibilities of interdisciplinary modeling and simulation [30]. These possibilities are briefly summarized in this context. The core component of the integrated simulation system ’MobileBody’ is the mechanical modelling environment a a M a aBILE ([22]). Its implementation using object oriented programming makes it easy to combine it with image processing code, visualization libraries and libraries for reading/writing files containing the model parameters. Furthermore, simple model components can be refined and replaced by more complex elements later. These may be individual models of anatomical joints of special interest or hand-tailored analysis modules for the clinical application. a a With M a aBILE, mechanical components are modeled as abstract mappings, termed kinetostatic transmission elements, which transmit motion and loads between sets of input and output variables called state objects. In this way, any mechanical model can be assembled by concatenating these kinetostatic transmission elements as are the parts of a a the real system. Since M a aBILE is implemented using the object-oriented programming language C++, new elements can be added easily using inheritance [31]. Mathematically, the operations related to the kinetostatic transmission elements correspond to well-known mappings of differential geometry: the transmission of position and velocity correspond to a nonlinear mapping between two smooth manifolds and the corresponding push-forward function for tangent vectors, respectively, while the force mapping corresponds to the pull-back function being applied to cotangent vectors. From the computational point of view, the concept renders a responsibility-driven client/server model [32] in which multibody operations are defined as “services” that an object provides at any time during program execution
87
88
Andres Kecskemethy / Procedia IUTAM 2 (2011) 86–92
Fig. 1: Functional principle of the simulation environment
independently of its internal implementation according to a specific “contract”. In the present mechanical modeling, the basic “contract” of kinetostatic transmission elements consists of two main services: one for transmission of motion and one for transmission of forces. Basically, most of the elements in the human gait model can be described by elementary objects such as joints, line segments for muscles, etc. More involved methods are required for the spine and the shoulder models, which display complex kinematical and soft-tissue interrelationships. In the following, we present, as an illustration of highercomplexity kinematical effects, a special foot model which allows for stable one-foot stance due to the modeling as circle-plane contacts. In forward dynamics simulation of biomechanical motion, the reaction forces between foot and floor are to be determined as a function of the foot penetration. Presented here is a simple foot model consisting of two rigid segments (fore foot and hind foot) connected by a revolute joint with a nonlinear spring-damper element in parallel. As shown in fig. 2, the input frame K of the foot element is fixed to the hind foot. Its origin is located at the center of the upper ankle joint, the z-axis is aligned with the axis of the ankle joint and the direction of the x-axis is chosen such that it is parallel to the ground in neutral null position. The revolute joint connecting the hind foot to the fore foot is aligned with the z-axis and its position relative to the input frame is described by the vector Δr 3 . The contact between the foot and the ground is modeled with two regularized circle-plane contact elements as described by [33]. One contact circle (radius R2 ) is fixed to the hind foot at position Δr2 parallel to the xz-plane of K. The other contact circle is fixed to the fore foot (radius R1 , offset Δr 1 ). The unknown parameters of the contact elements, of the inter-segment revolute joint stiffness and the offset vectors Δr1 , Δr 2 and Δr 3 are identified using measured contact forces and foot positions. From this measured data, the wrench wa (ti ) at the ankle joint is computed at m discrete points in time ti . The ankle joint wrench is applied to the input frame
89
Andres Kecskemethy / Procedia IUTAM 2 (2011) 86–92
y z
joint axis x
hindfoot contact cylinder
forefoot contact cylinder ground plane y z
K Δr2 x
Δr3
R2
k,c +
Δr1 R1
hindfoot contact circle ϕ
-
intersegment joint with elastic element forefoot contact circle ground plane
Fig. 2: Elastokinematic foot model with circle-plane contact pairs
K of the foot element and the static equilibrium pose of the foot model under this load is determined for i = 1, . . . , m. In turn, the pose at static equilibrium is parameterized by the global coordinates rsim,i of the origin of frame K and the Bryant angles p of K and compared to the measured pose parameterized in the same way (rmeas (ti ) and p (ti )). sim,i meas Parameter estimation is performed by minimizing the least squares cost function 1 (r (ti ) − r sim,i )2 + (p (ti ) − p )2 meas sim,i 2 i=1 meas m
F(x) =
(1)
where x is the vector of unknown parameters of the foot model. Fig. 3 shows the measured angle between hindfoot measured model
35 30
rotation about z-axis in ◦
25 20 15 10 5 0 −5 −10 3.8
3.9
4
4.1
4.2
4.3
4.4
t in seconds
Fig. 3: Measured and simulated orientation of hind foot with identified parameters
and ground and the simulated angle obtained for the identified parameters, which lead to acceptable differences of maximally 5◦ . The described multibody models were incorporated in a modeling and simulation environment called ’MobileBody’ that provides the basic features required for biomechanical gait analysis and can be extended both for dynamical issues and for specific modules that focus on clinical applications. The emphasis of the program is to supply
90
Andres Kecskemethy / Procedia IUTAM 2 (2011) 86–92
a highly-efficient code paired with an intuitive interface such as to be able to support the doctors in planning and assessment of medical treatments, for which an intuitive user interface was developed. 3. Results Based on this modeling and simulation environment, it is possible to perform inverse and direct dynamic analysis, as well as to implement additional features such as augmented-reality 3D motion visualization and analysis for patient-specific gait diagnosis (fig. 4). These features are believed to become of higher importance when medical personnel will use the tools, as they will require a more intuitive human-machine interface than today’s more numerical view. Tools that are of importance in this setting are the data fusion of MRI and motion capturing for better patientspecific data ([30]), composite numerical clinical scoring algorithms ([30]), optimization-based muscle recruitment identification ([20]) and verification by interval analysis [34]. Based on these methods, patient-specific numerical medical diagnosis might become possible at the doctor’s desk, which is a goal which might be reached in the near future. patient-specific bone geometries (MRI und X-Ray) XML
Motion analysis system: • Vicon Nexus
icon Nexus Vicon
Augmented-Reality Server • biomechanical prognosis system • Studierstube visualization library
marker positions ( patient)
VICON
marker trajectories (Patient) & jointaxes and jointangles
PC
position und rotation HMD
IV, WRL, STL
AR Server
marker positions (HMD)
3Dbone model Vicon Kameras Tracking
Tracking
virtual overlay of patient and bone model
Patient mit Markern
Arzt mit HMD
Fig. 4: Augmented reality environment with virtual X-Ray functionality
4. Discussion By the use of object-oriented paradigm, new possibilities arise for (1) easily exchanging biomechanical effects from simple to complex in a running biomechanical model, (2) intuitive user interfaces and (3) embedding of biomechanical issues in interdisciplinary applications. Such aspects might become important when pursuing a medical tool suitable for planning and assessment of operations and rehabilitation measures of the human musculoskeletal system in clinical practice. 5. Acknowledgement Funding by European Union (European regional development fund ERDF) under initiative of the Ministry of Innovation, Science and Research of the State of Northern Rhine-Westfalia, project number 005-0608-0004 is gratefully acknowledged.
Andres Kecskemethy / Procedia IUTAM 2 (2011) 86–92
91
References [1] D. Winter, R. Greenlaw, D. Hobson, Television-Computer Analysis of Kinematics of Human Gait, Computers and Biomedical Research 5 (5) (1972) 498–504. [2] A. Cappozzo, T. Leo, A. Pedotti, A general computing method for the analysis of human locomotion, Journal of Biomechanics 8 (5) (1975) 307–320. [3] M. Whittle, Gait Analysis. An introduction, Butterworth Heinemann, 1991. [4] V. Saraph, E. Zwick, G. Steinwender, C. Auner, F. Schneider, W. Linhart, Leg lengthening as part of gait improvement surgery in cerebral palsy: an evaluation using gait analysis, Gait and Posture 23 (1) (2006) 83–90. [5] F. Anderson, M. Pandy, A Dynamic Optimization Solution for Vertical Jumping in Three Dimensions, Computer Methods in Biomechanics and Biomedical Engineering 2 (1999) 201–231. [6] G. Yamaguchi, Dynamic Modeling of Musculoskeletal Motion, Kluwer Academic Publishers, Norwell, Massachusetts, 2001. [7] A. Veloso, G. Esteves, S. Silva, C. Ferreira, F. Branda˜o, Biomechanics modeling of human musculoskeletal system using adams multibody dynamics package, in: Proceedings of the 24th IASTED international conference on Biomedical engineering, Innsbruck, Austria, 2006, pp. 401–407. [8] M. Lengsfeld, T. Pressel, U. Stammberger, Lengths and lever arms of hip joint muscles: geometrical analyses using a human multibody model, Gait and Posture 6 (1997) 18–26. [9] S. Delp, K. Loan, A computational framework for simulating and analyzing human andanimal movement, Computing in Science & Engineering 1 (2000) 46–55. [10] M. Damsgaard, J. Rasmussen, S. Christensen, E. Surma, M. de Zee, Analysis of musculoskeletal systems in the anybody modeling system, Simulation Modelling Practice and Theory 14 (2006) 1100–1111. [11] M. Kadaba, H. Ramakrishnan, M. Wootten, Measurement of lower extremity kinematics during level walking, Journal of Orthopaedic Research 8 (3) (1990) 383–392. ˜ [12] R. I. Davis, S. Ounpuu, D. Tyburski, J. Gage, A gait analysis data collection and reduction technique, Human Motion Science 10 (5) (1991) 575–587. [13] C. Vaughan, B. Davis, J. O’Connor, Dynamics of Human Gait, 2nd Edition, Kiboho Publishers, Cape Town, South Africa, 1992. [14] E. Garling, B. Kaptein, B. Mertens, W. Barendregt, H. Veeger, R. Nelissen, E. Valstar, Soft-tissue artefact assessment during step-up using fluoroscopy and skin-mounted markers, Journal of Biomechanics 40 (Supplement 1) (2007) S18–S24. [15] A. Leardini, L. Chiari, U. Della Groce, A. Cappozzo, Human movement analysis using stereophotogrammetry Part 3. Soft tissue artifact assessment and compensation, Gait and Posture 21 (2) (2005) 212–225. [16] A. Ferrari, M. Benedetti, E. Pavan, C. Frigo, D. Bettinelli, M. Rabuffetti, P. Crenna, A. Leardini, Quantitative comparison of five current protocols in gait analysis, Gait and PostureDoi:10.1016/j.gaitpost.2007.11.009. [17] M. Klein Horsman, H. Koopman, F. van der Helm, L. Poliacu Pros´e, H. Veeger, Morphological muscle and joint parameters for musculoskeletal modelling of the lower extremity, Clinical Biomechanics 22 (2) (2007) 239–247. [18] J. Bauer, M. Pavol, C. Snow, W. Hayes, MRI-derived body segment parameters of children differ from age-based estimates derived using photogrammetry, Journal of Biomechanics 40 (2007) 2904–2910. [19] M. Daniel, A. Igliˇc, V. Kralj-Igliˇc, S. Konviˇckov´a, Computer system for definition of the quantitative geometry of musculature from ct images, Computer Methods in Biomechanics and Biomedical Engineering 8 (2005) 25–29. [20] J. Ambrosio, A. Kecskem´ethy, Multibody dynamics of biomechanical models for human motion via optimization, in: J. C. G. Orden, J. M. Goicolea, J. Cuadrado (Eds.), Computational Methods in Applied Sciences, no. 4, Springer Netherlands, 2007, pp. 245 – 272. [21] M. Praagmana, E. Chadwick, F. van der Helm, H. Veegera, The relationship between two different mechanical cost functions and muscle oxygen consumption, Journal of Biomechanics 39 (4) (2006) 758–765. [22] A. Kecskem´ethy, M. Hiller, An object-oriented approach for an effective formulation of multibody dynamics, Computer Methods in Applied Mechanics and Engineering 115 (1994) 287–314. [23] A. Kecskem´ethy, I. Maˇsi´c, D. Raab, M. T¨andl, Workspace fitting and control for a serial-robot motion simulator, in: Proceedings of the 2nd European Conference on Mechanism Science, Cassino, Italy, 2008, pp. 183–190. [24] F. Geu Flores, A. Kecskem´ethy, A. P¨ottker, Workspace analysis and maximal force calculation of a face-shovel excavator using kinematical transformers, in: CD Proceedings of the 12th World Congress in Mechanism and Machine Science, Besanc¸on, France, 2007. [25] A. Kecskem´ethy, M. T¨andl, A robust model for 3D tracking in object-oriented multibody systems based on singularity-free Frenet framing, in: Proceedings of the 10th International Symposium on Advances in Robot Kinematics, Ljubljana, Slovenia, 2006, pp. 255–264. [26] R. Cuypers, Z. Tang, W. Luther, J. Pauli, Efficient and Accurate Femur Reconstruction using Model-Based Segmentation and Superquadric Shapes, in: R. Merrell, R. Cooper (Eds.), Proceedings of the IASTED International Conference on Telehealth and Assistive Technologies, ACTA Press, Anaheim, USA, 2008, pp. 99–104. [27] D. Strobach, A. Kecskem´ethy, E. Auer, W. Luther, G. Steinwender, B. Zwick, A sensitivity analysis of origin and insertion points of Hill muscle models with respect to gait dynamics, in: CD Proceedings of the ECCOMAS Thematic Conference on Advances in Computational Multibody Dynamics, Milano, Italy, 2007. [28] X. Liu, A. Kecskem´ethy, M. T¨andl, A self-stabilized foot-ground contact model using two segments and cylinder-plane pairs, in: Proceedings of the first i-FAB congress, Bologna, Italy, 2008. [29] E. Auer, A. Rauh, E. Hofer, W. Luther, Validated modeling of mechanical systems with smartmobile: Improvement of performance by valencia-ivp, in: Reliable Implementation of Real Number Algorithms: Theory and Practice, Springer Netherlands, 2008. [30] M. T¨andl, T. Stark, N. Erol, F. L¨oer, A. Kecskem´ethy, An object-oriented approach for simulation of human gait motion based on motion tracking, International Journal of Applied Mathematics and Computer Science (AMCS) 19 (3) (2009) 469–483. [31] B. Stroustrup, The C++ Programming Language, second edition, Addison-Wesley Series in Computer Science, Addison-Wesley Publishing Company, Reading, MA, 1991. [32] R. Wirfs-Brock, B. Wilkerson, Object-oriented design: A responsibility-driven approach, in: OOPSLA ’89 Proceedings, 1989, pp. 71–75.
92
Andres Kecskemethy / Procedia IUTAM 2 (2011) 86–92
[33] A. Kecskem´ethy, C. Lange, G. Grabner, A geometric model for cylinder-cylinder impact with application to vertebrae motion simulation, in: J. Lenarˇciˇc, M. M. Staniˇsi´c (Eds.), 7th International Symposium on Advances in Robot Kinematics, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000, Piran-Portoroz, Slovenia, 2000, pp. 345–354. [34] A. Cuyt (Ed.), Numerical Verification Assessment in Computational Biomechanics, Vol. 5492, 2009.
