Musculoskeletal Modeling and Physiological Validation

Akihiko Murai1, Kazunari Takeichi1, Taira Miyatake1, and Yoshihiko Nakamura1

Abstract— Digital human models are applied in human mo-tion analysis and simulation. They can be applied in rehabil-itation, sports science, biomedicine, and so on. The difficultyof validation of models and analysis algorithms constricts theirpractical usage. In this paper, we show our work on humanmotion analysis using a whole-body musculoskeletal model andvalidate the analysis results in terms of physiology. First, webuild the detailed musculoskeletal model that represents thekinematics and dynamics characteristics of human body. Opti-cal motion capture, force plates and electromyography(EMG)are used for human motion capture. We estimate the muscletensions required to generate the captured motion sequencebased on an inverse kinematics and dynamics computation anda mathematical optimization. The estimated muscle tensions forlocomotion cycles are compared with the tensions computedfrom the simultaneously measured EMG data and a physiolog-ical muscle model. The model and the analysis algorithm arealso applied to a neurophysiological phenomenon, the nontrivialpreferred direction that is a result of the cosine tuning. Ourmodel and analysis algorithm achieves results that correspondwith the experimental physiological data well. The possibleapplications of our model and algorithm involve rehabilitation,sports science, biomedicine, and robotics, e.g. a controller of anexoskeletal robot for human support.

I. INTRODUCTION

Many digital human models have been developed andapplied mainly to human motion analysis and simulation.They are applied to the medical diagnosis, the rehabilitation,the sports training and so on in the research fields. Afew of them are utilized for the car design, the sportingequipment development and so on in the practical usage.The advantages of using these digital human models aremainly money and time. The simulations of human motionswith products, e.g. boarding a car, sitting on a chair, can berealized with various parameters in the human and productside. The analysis results give ergonomics insights intoproduct design. However they currently do not come intowidespread use in the practical usage. One of the reasonsare the difficulty of validating the models and the analysisalgorithms. The models have a huge numbers of elements,degrees of freedom, and parameters and they differ fromperson to person. CT and MRI scanning provide detailed andsubject specific data, but are difficult to apply to individualsfrom the cost and the ethical points of view. The analysisresults, e.g. muscle tensions, joint loads, are technically andethically difficult to measure directly and experimentallyfrom the real human. These ambiguities prevent widespreadpractical usage of models and analysis algorithms.

1Akihiko Murai, Kazunari Takeichi, Taira Miyatake, and Yoshihiko Naka-mura are with Graduate School of Information Science and Technology,The Department of Mechano-Informatics, The University of Tokyo, 7-3-1Hongo, Bunkyo-ku, Tokyo, Japan murai@ynl.t.u-tokyo.ac.jp

The authors have been developing the musculoskeletalmodel and the motion analysis algorithm. The musculoskele-tal model [1], [2], [3] represents the kinematics and dynamicscharacteristics of human whole-body. At the motion analysis,firstly the human motion sequence is captured by an opticalmotion capture system, force plates, and electromyogra-phy(EMG). The inverse kinematics computation based onthe kinematic properties of the skeleton model providesjoint angle data. The inverse dynamics computation basedon the computed joint angle data and the inertia propertiesof the skeleton model provides contact forces between thehuman and the environment and generalized forces, thatis the joint torques in this case. Lastly joint torques aredistributed to muscle tensions according to a mathematicaloptimization with dynamics constraints and physiologicalevaluation functions.

Human motion measurements are strictly limited from thetechnological and ethical point of view and this makes thevalidations of models and analysis algorithms difficult. Thegeneral methods are with the optical or mechanical motioncapture system, the force sensors, the EMG, the heart ratemonitors and so on. At the analysis, all the available dataare used to improve the analysis accuracy in the frameworksof e.g. mathematical optimization. Cross-validation can beapplied for the validation by estimating the measurable dataand comparing the estimated results with the experimentallymeasured data. In our setting, the measurable data correspondto EMG data. Muscle tensions are estimated without usingthe measured EMG data and compared with the muscletensions that are computed based on the EMG data and thephysiological muscle model in this paper.

Our second validation criterion is the comparison withneurophysiological phenomena. There are phenomena thatcannot be explained by only the dynamics constraints, e.g.signal-dependent noise or nontrivial preferred direction [4],[5], [6], [7], [8]. The realization of these phenomena basedon the model and the analysis algorithm increases the validityof the mathematical implementation of the physiologicalsystem. The preferred direction is simulated and comparedwith the experimental data in this paper.

The rest of this paper is organized as follows. In Section II,our musculoskeletal model is described. The computationalalgorithm for estimating the muscle tensions is shown inSection III. The estimated muscle tensions are validated inthe physiological point of view in Section IV and Section V,followed by the concluding remarks.

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Fig. 1. The musculoskeletal human model.

II. MUSCULOSKELETAL MODEL

The detailed whole-body human musculoskeletalmodel(Fig.1) consists of the skeleton model and themusculo-tendon network. We scan the skeletal preparationby Computer Tomography(CT) at the University of Tokyoand generate the Digital Imaging and Communication inMedicine(DICOM) data. The post process, e.g. polygonrendering, noise reduction, segmentation, is applied to thedata and the polygon shape model is completed. The modelis published on the website [9]. The model becomes aset of rigid linkages that have the inertia properties of theaverage male adult, whose weight is 68 [kg]. Each linkageis connected to each other by a mechanical joint, usuallyspherical, and the whole-model consists of 55 linkages andhas 155 degrees of freedom.

The musculo-tendon network consists of muscles, tendons,ligaments, and cartilages. The muscles work as actuators andthe others work as kinematic constraints between linkages.All elements are implemented as the ideal wire modelsthat are represented by the origin, end, and via points thatare fixed on the linkages. We also introduce the idea ofthe virtual link that is a massless linkage to represent abranch of elements e.g. the Achilles tendon, Soleus, andGastrocnemius [10]. Our musculo-tendon network consistsof 989 muscles, 50 tendons, 117 ligaments, and 34 cartilagesthat appear in [11], [12]. The kinematics and dynamics ofthe human body are computed based on this musculoskeletalmodel and the algorithm that is mentioned in the followingsection.

III. COMPUTATIONAL ALGORITHM FOR ESTIMATION OFMUSCLE TENSION

The analysis algorithm to estimate the muscle tensionsfrom the captured motion data are shown in this section.Firstly, the inverse kinematics computation with the captured3D positional marker data and the skeleton model allows usto compute the joint angle data for each frame of the capturedmotion sequence. Then, the inverse dynamics computationwith the motion data and the inertia parameters of theskeleton model allows us to estimate the external forcesapplied to the human (the contact forces) and the generalizedforces (the joint torques) for each frame of the capturedmotion sequence.

Finally, the joint torques τ are distributed to the muscletensions f based on the kinematic muscle pathway on themusculoskeletal model. The relationship between the jointtorques and the muscle tensions are as follows:

Ji,j(θ) =∂lj∂θi

(1)

τ = JT (θ)f (2)f = (JT (θ))#τ , (3)

where Ji,j(θ) represents the Jacobian matrix of the j-thmuscle length w.r.t. the i-th joint angle. The matrix JT isequal to the moment arm of j-th muscle w.r.t. i-th joint,and multiplying its inverse matrix to the joint torques allowsus to compute the muscle tensions. Here, the human issignificantly redundantly actuated by the muscles, in which155 DOF are actuated by around 1000 muscles, and thejoint torques cannot be identically distributed to the muscletensions only based on the dynamics constraints. In thefields of physiology and computational neurophysiology, theinternal models for motor control [13], the minimizationof the motor command signal [14], the minimization ofmotion jerk [15], the signal dependent noise [4] etc. havebeen proposed as the controller with which human controltheir body. We adopt the minimization of the square sum ofmuscle tensions that is a common way to represent the humancontroller and implement that in the form of a mathematicaloptimization problem.

Z = δTτ aTτ δτ + δTf a

Tf δf + δTmaT

mδm (4)

δτ = τ − JTf (5)δf = f − f∗, f ≤ 0 (6)

δm = EGf , (7)

where the first term minimizes error between the torques thatis realized by the estimated muscle tensions and the onesthat are required to realize the captured motion sequence.The second term minimizes the error between the estimatedmuscle tensions and the ones computed from the measuredEMG data and the physiological muscle model [16] for themuscles whose EMG data are measured. The activities of theco-contracted muscles that cannot be estimated only fromthe motion data can be estimated from this term. f∗ = 0for the muscles whose EMG are not measured and this termminimizes the quadratic sum of the muscle tensions. Thethird term minimizes the variance of the muscles tensionsthat belong to the same synergetic muscle group and satisfythe physiological correspondence. This algorithm allowsus to compute reasonable muscle tensions that satisfy thedynamics constraint as well as the physiological constraintby considering them in the framework of the mathematicaloptimization. The computed muscle tensions are validated inthe next section.

The real-time system, musculoskeletal-see-through mir-ror(Fig.2) [17] is developed with our musculoskeletal modeland the muscle tension estimation algorithm. The muscletensions are estimated from the data captured by the opticalmotion capture system, the force plates, and EMG in real

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Mocap System (Motion Analysis)

Video Camera

Wireless EMG (DELSYS)

Display

Force Plate (KISTLER)

PC for System Control

Fig. 2. Overview of the real-time muscle tension visualization system.

Fig. 3. Images captured from the musculoskeletal-see-through mirror.

time. The estimated muscle tensions are visualized as themusculoskeletal model and the rendered musculoskeletalmodel is overlayed on top of the real-time video image ofthe subject. The subject feels that he/she sees his/her muscu-loskeletal system through the cloth and skin(Fig.3). The newalgorithm that computes the reasonable approximate muscletensions based on the neural connection between musclesknown as synergetic muscle group, and the algorithm canestimate the tensions of 274 muscles in only 16ms, and thetotal system refreshes at about 15 fps [17]. The speeding upof the inverse kinematics computation and the mathematicaloptimization realizes the estimation of the tensions of 989muscles in real time [18]. Possible applications are the bio-feedback system for assisting rehabilitation or sports training.

IV. PHYSIOLOGICAL VALIDATION WITH EMG DATA

The validations are absolutely important to put this modeland computational algorithm to practical use. The experimen-tal direct measurement of the muscle tensions is significantlydifficult from the technical and ethical point of view. TheEMG data are often used as the data that represent the muscleactivities directly. In the previous section, the algorithmto estimate the muscles tensions uses the EMG data asreference to increase the estimation accuracy. In this section,we validate our model and algorithm by comparing:

1) the muscle tensions estimated from the inverse kine-matics and dynamics computation and the mathemati-cal optimization, which does not use the EMG data asthe reference (namely, f∗ = 0 for all muscles), and

TABLE IERRORS OF ESTIMATED MUSCLE TENSIONS

Name Right Error [%] Left Error [%]Rectus Femoris 6.73E+01 1.46E+02Vastus Lateralis 1.10E+01 8.89E+00

Biceps Femoris Caput Longum 5.90E+01 8.52E+01Biceps Femoris Caput Breve 4.48E+02 3.88E+02

Tibialis Anterior 2.02E+02 3.60E+01Gastrocnemius 2.15E+01 3.23E+01

Soleus 9.25E+00 8.45E+00

2) the muscle tensions estimated from the EMG dataand the Hill-type muscle model [16], that computesmuscle tension from its length, velocity, and maximumisometric tension.

The locomotion cycles are captured by the commercialoptical motion capture system with 12 cameras (MotionAnalysis Raptor system, resolution: 2352x1278 [pixels],frame rate: 200 [fps]). The subject wears 35 markers whoselocations are determined based on the improved version ofthe Helen Hayes Hospital marker set. The EMG data ofright/left Rectus Femoris, Vastus Lateralis, Biceps FemorisCaput Longum, Biceps Femoris Caput Breve, Tibialis Ante-rior, Gastrocnemius, and Soleus are simultaneously recordedusing Delsys Trigno EMG system with 14 pairs of electrodesat a rate of 1 [kHz]. The EMG data are post-processed bymean subtraction, rectification, and a Butterworth bandpassfilter with a cut-off frequency of 10-100 [Hz]. The contactforces between the subject and the floor are also recordedusing three KISTLER force plates, each of them can measurethe 6 axis contact force and momentum at a rate of 1 [kHz].Fig.4 shows one of the locomotion cycle that is used for thevalidation.

Figs.5-11 show the results of the validation. Each graphsrepresents the tensions of the right leg muscle that arecomputed by 1) and 2) algorithms. The horizontal axisrepresents the time that is normalized by the locomotioncycle. Here, 0 and 1 represent the right heel strike timing.The vertical axis represents the muscle tension [N]. The redline represents the tension computed by 1) and the bluedline represents the tension computed by 2). 20 locomotioncycles are used for this validation. The thick line representstheir average and the translucent area represents the standarddeviation. The errors of the tensions computed by 1) w.r.t.the tensions computed by 2) are shown in Table I. The detailsare discussed in section VI.

V. CONSIDERATION BASED ON COSINE TUNING

Our model and algorithm to estimate the muscle tensionsare also validated based on the neurophysiological experi-ment. Most of the musculoskeletal phenomena that happenon the human body cannot be described only from thedynamics constraints because the human body is significantlyredundantly actuated [19]. One of the phenomena is thecosine tuning [5], [6], [7], [8] that shows how our centralnerve system selects a unique muscle activity pattern withthe redundantly actuated system coming from the existence

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Fig. 4. Locomotion cycle used for the validation. Muscle color changes from green to red when it is activated.

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Fig. 5. Tensions of right Rectus Femoris. Red line: estimated bythe dynamics computation and mathematical optimization and blue line:estimated by the EMG data and the physiological muscle model.

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Fig. 6. Tensions of right Vastus Lateralis. Red line: estimated bythe dynamics computation and mathematical optimization and blue line:estimated by the EMG data and the physiological muscle model.

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Fig. 7. Tensions of right Biceps Femoris Caput Longum. Red line:estimated by the dynamics computation and mathematical optimization andblue line: estimated by the EMG data and the physiological muscle model.

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EMG and muscle model

Fig. 8. Tensions of right Biceps Femoris Caput Breve. Red line: estimatedby the dynamics computation and mathematical optimization and blue line:estimated by the EMG data and the physiological muscle model.

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Fig. 9. Tensions of right Tibialis Anterior. Red line: estimated bythe dynamics computation and mathematical optimization and blue line:estimated by the EMG data and the physiological muscle model.

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EMG and muscle model

Fig. 10. Tensions of right Gastrocnemius. Red line: estimated by the dy-namics computation and mathematical optimization and blue line: estimatedby the EMG data and the physiological muscle model.

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Fig. 11. Tensions of right Soleus. Red line: estimated by the dynamicscomputation and mathematical optimization and blue line: estimated by theEMG data and the physiological muscle model.

of bi-articular muscles. This phenomenon results in thenontrivial preferred direction that is defined as the directionwhere the muscle is activated maximally on the torqueplane is different from its mechanical pulling direction. Wesimulate the muscle activity in the case that is experimentallymeasured in the literature and compare the results with [8].

The musculoskeletal model is set to the posture that isrequired from the subject in [8], in which the hip joint flexes90 [degree] and the knee joint flexes 90 [degree]. The 6DOF of the hip joint are fixed to the space and the variouscombinations of torques are applied to the hip and the kneejoints. The inverse dynamics computation and the mathemati-cal optimization provide the muscle activities for each torquecombination. The results are shown in Figs.12-14. Fig.12shows the simulated muscle tensions for the various com-binations of the hip and knee joint torques. The horizontalaxes represent the hip and knee torque [Nm] respectively andthe vertical axis represents the muscle tension [N]. Figs.13-14 represent the preferred directions of the muscles. Thehorizontal axis represents the knee torque and the verticalaxis represents the hip torque. Each arrow represents thepreferred direction of each muscle respectively. In Fig.13, theblack arrow represents the simulated preferred direction andthe gray dashed arrow represents the one measured in [8]. InFig.14, the black arrow represents the simulated preferreddirections when Rectus Femoris is missing and the graydashed arrow represents the simulated preferred directionswith all muscles.

VI. DISCUSSION

We can observe the following points in the experimentaland simulation results:

1) The inverse dynamics computation and the mathemat-ical optimization that minimizes the quadratic sum ofmuscle tensions estimates the tensions well for Vas-tus Lateralis, Gastrocnemius, and Soleus with errorsaround 10 %. The standard deviations of the tensions in20 locomotion cycles are also small for those muscles.

2) The errors are significantly large for Biceps FemorisCaput Breve and Tibialis Anterior with errors around

VM

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Fig. 12. Simulated muscle tensions for various combinations of hip andknee joint torques. RF: Rectus Femoris, VM: Vastus Medialis, BFL: BicepsFemoris Caput Longum, ST: Semitendinosus, and GM: Gluteus Maximus.

300 %, though the activation patterns and timing arewell estimated for these muscles.

3) The preferred directions of the leg muscles w.r.t. thehip and knee joint torques are well simulated andcorrespond to the ones measured in [8].

4) The preferred direction deviates to fill the space ofRectus Femoris when Rectus Femoris is missing andthat corresponds with the literature.

These results have the following implications:

1) The inverse dynamics computation and the mathemat-ical optimization that minimize the quadratic sum ofmuscle tensions estimates the activation pattern andtiming precisely, though there are large errors in theamplitude in some muscle tensions. The possible appli-cations involve a controller of an exoskeletal robot forhuman support in which sensing the muscle activationtiming is important.

2) Large errors in the estimated tensions of BicepsFemoris Caput Breve and Tibialis Anterior come fromthe individual variations of the parameters, e.g. theisometric maximum force and the muscle fiber length.The similarity of the activation timings and the wavepatterns implies the possibility of the parameters iden-tifications from these errors.

3) The simulated preferred directions correspond with themeasured ones well. This result shows that the mini-mization of the quadratic sum of the muscle tensions

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BFS

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HipTorque

Fig. 13. Simulated preferred direction for the hip and knee joint torquesand the measured ones from [8]. RF: Rectus Femoris, VL: Vastus Lateralis,VM: Vastus Medialis, GM: Gluteus Maximus, BFL: Biceps Femoris CaputLongum, BFS: Biceps Femoris Caput Breve, ST: Semitendinosus, and GAS:Gastrocnemius.

BFS

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Fig. 14. Simulated preferred direction for the hip and knee joint torquesin the case that Rectus Femoris is removed.

represents how our central nerve system selects theunique muscle activity pattern.

Several directions remain for future works. Our goal isto put the musculoskeletal model and the muscle tensionestimation algorithm to practical use. Our musculoskeletalmodel is currently not individualized for the muscle parame-ters and this causes large errors in muscle tension estimation.We expect that the implementation of the muscle parameterindividualization process with comparison between the sim-ulated preferred directions and the experimentally measuredones would significantly improve the accuracy of the muscletension estimation. Another interesting direction is to applythe muscle tension estimation algorithm to nursing care. Ouralgorithm can estimate the muscle tensions when a certainmuscle is missing. This estimation algorithm would work forcomputing the necessary support for a patient with missingor weakened muscles.

ACKNOWLEDGEMENT

This research is supported by HPCI STRATEGIC PRO-GRAM, Computational Life Science and Application in

Drug Discovery and Medical Development, MEXT, Japan.The authors gratefully acknowledge the support by Grant-in-Aid for Young Scientists (B) #25730156.

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