Acta of Bioengineering and BiomechanicsVol. 10, No. 1, 2008

Kinematical analysis of mandibular motionin a sagittal plane

J. MARGIELEWICZ1, W. CHLADEK2, *, T. LIPSKI3

1, 2 Department of Medical Engineering, Silesian Technical University, Poland.3 Chair of Department of Dentistry Prosthetics,

Silesian Medical University, Poland.

The paper presents a kinematical model enabling the analysis of mandibular motion in a sagittal plane. Based on the recorded tra-jectories of incisors, the configuration coordinates were identified. The configuration coordinates explicitly identify the position andorientation of the mandible during motion. Such values are basic to the evaluation of alteration in muscle length and to the orientation offorces in particular muscles. This paper also deals with the influence of the coefficients of the weight matrix on the character of the solu-tions of the configuration coordinates applied in the model study of the kinematical chain. The results of the numerical calculations ob-tained demonstrated that the trajectory representation was in a considerable concordance with the data recorded.

Key words: biomechanics, modelling, trajectory, stomatognathic system

1. Introduction

Mandibular motions against a static maxillary boneand the state of dental arches are responsible for a properspeech and mastication processes. These functions arecontrolled by the central nervous system, providingguidance according to specific memorized codes ofpractice. A mandible is attached to the skull with a hingejoint and can move up and down within the range deter-mined by the anatomy of the facial skeleton [1]. Thediagnostic records of mandible motions, defined asfunctional registrations, are one of the most crucial fac-tors determining the choice of treatment method in thedysfunctions of a stomatognathic system [6]. The com-bination of mandibular motions with the correspondingposition of dental arches during prosthetic reconstruc-tions poses a separate problem. The aforesaid problemshave considerable importance, therefore the studies onthe model analysis of mandible motions have been un-dertaken. They can provide the results useful both for the

estimation of dental arch loads and for new designs ofarticulators. Based on kinematical relationships describ-ing the nature of mandible motion it is possible to iden-tify motion parameters. These relationships serve thispurpose in the cognitive apparatus of mechanics. In or-der to describe positions, speeds and accelerations, it isnecessary to make use of a mathematical apparatus effi-cient in numerical calculations. In this connection, inorder to define the positions and the orientation of themandible in selected stages of motion, one requires theinformation on configuration coordinates in these stages.Such information can be gained from clinical trials car-ried out, for instance, by applying computerized axialtomography. A complete image of mandible behaviourin motion is obtained as a result. However, sucha method is very expensive and it is not recommended innumerous cases. Another method providing informationon mandibular kinematics is plotting the trajectory of itscharacteristic points on plates by using a scriber. Sucha method is less precise, but non-invasive and inexpen-sive in use. The precision of the methods with articula-

______________________________

* Corresponding author: Wiesław Chladek, Research Group for Medical Engineering, Silesian Technical University, ul. Krasiń-skiego 8, 40-019 Katowice, Poland. E-mail: chladekw@interia.pl

Received: May 23, 2007Accepted for publication: October 16, 2007

J. MARGIELEWICZ et al.10

tors increases together with the experience of a re-searcher [13]. The data recorded should be further proc-essed, regardless of the diagnostic devices applied, inorder to carry out a kinematical analysis and should beadjusted to a numerical experiment. Only the combina-tion of clinical trial and numerical study can provide uswith comprehensive information on a functional behav-iour of mandible. Mandibular kinematics can be consid-ered in two ways [3], [4], [7]. The first one is by plottingthe trajectory for selected points at given configurationcoordinates (direct kinematics task). The second wayconsists in the plotting of configuration coordinatesbased on a given trajectory (inverse kinematics task).The configuration of the kinematical chain can be identi-fied by means of several methods, i.e., analytical meth-ods, numerical methods or neural networks. Each of theabove-mentioned methods has its advantages and disad-vantages. The major disadvantage of neural networks isthe need for the preparation of training data. An appro-priate preparation of such data is a time-consuming andlaborious process, and the efficiency of identification ofthe kinematical chain depends substantially on the neuralnetwork ‘training’. Therefore, the application of neuralnetworks in the diagnostics of mandible kinematics isconsidered to be inefficient. The solutions derived fromanalytical method provide the most substantial benefits.However, these methods have also some disadvantages.The main one is the lack of the possibility of obtainingunique explicit solutions. Despite a considerably longtime required for calculations compared with analyticalmethods, this drawback does not appear to be highlyimportant in mandibular kinematics. Therefore, themodel study of mandibular kinematics demonstrated inthis paper applied the aforesaid numerical methods.

2. Biomechanics ofmandibular motions

The range of mandibular motions is determined bythe surfaces of temoporomandibular joints, ligaments,superficial tooth structure and by the spatial structureof the muscular system. There are two basic types ofmotion [5], [8], [13]. The first type includes articula-tion motions which occur in condition of occlusalcontacts. The second one includes the so-called freemandibular motions performed when occlusal contactdoes not occur. These motions result from a synchro-nised activity of neuromuscular system. Numericalstudies of mandibular kinematics are carried out basedon the recorded trajectories of its selected points. The

characteristic points most frequently chosen are thoseplaced between the two bottom incisors, i.e., the inci-sion point and points located in the surrounding areaof the centre of the mandible head. An exact locationof the characteristic point enables the definition ofmandible orientation in selected time intervals. Duringthe analysis of recorded trajectories the limiting posi-tions, named border positions, are observed. Thecharacteristic feature of such positions is their repeat-ability. The repeatability of limiting trajectoriesproves to be one of the criteria fundamental to theapplication of a mathematical model in clinical diag-nostics. The mandible, when dislocating within themaximum achieveable range of positions, followsa path defined as a limiting trajectory. The limitingtrajectory defines a working space within which freeand functional motions are performed. The recordedtrajectories of free and functional motions, in contrastwith the limiting motions, are not required to be re-peatable. Generally speaking, there are three basicmotions performed in the temporomandibular joints:

• Abduction and elevation.• Protraction and retraction.• Rotations in a horizontal plane.

Fig. 1. Model of mandibular head position in a socketduring abduction and adduction [5]

Abduction or elevation is performed simultane-ously in both joints. The axis intersecting the mandi-ble heads, against which abduction or elevation is

Kinematical analysis of mandibular motion in a sagittal plane 11

performed, is not a fixed axis. The axis of rotationintersects the mandible heads and is located in closeproximity to their central points. Concurrently witha gradual increase in mandibular opening, the axisdislocates in anterior direction. Additionally, the mo-tion of mandible heads in temporomandibular joints isconsiderably influenced by the joint geometry as wellas by the susceptibility of intra-articular discs. Themodel of mandibular head position in a socket duringabduction and adduction is demonstrated in figure 1.

During mandible abduction the intra-articular discsslide in a socket in anterior direction. Concurrently,the mandible heads are dislocated towards the inferiorsurface of the articular tubercle. Such a motion is de-fined in specialist literature as rotary sliding. Therange of dislocations of mandible heads is limited bythe anatomy of joints, ligaments and joint capsules.The element responsible for mandible abduction isa suprahyoid muscle which, when the hyoid bone isimmobilized, causes its abduction. Abduction is en-hanced by gravitational force. For the elevation andocclusion of dental arches the temporomandibularmuscles are of primary importance. Essential charac-teristics of mandible heads and the mandible itself areobtained as a result of applying dental diagnostic de-vices. Such devices allow the mandibular motions tobe recorded in three planes. The recording of mandi-ble trajectory in the sagittal plane is performed in or-der to identify the following basic stages of motion:

• Limiting motion during posterior opening.• Limiting motion during anterior opening.• Limiting motion during mandible protraction

with occlusal contacts.

Fig. 2. Mandibular trajectory in the sagittal plane

Limiting motion during posterior opening is per-formed in two stages. In the first stage, the mandibleheads are in fovea articularis. In such a case, the ab-duction of the mandible can be sufficiently preciselydescribed by rotary motion (figure 2, curve 1), and the

rotation axis crosses the central points of the heads.The actual shape of the trajectory plotted by the inci-sion point of the mandible differs from the shapedemonstrated in figure 2. The trajectory represents anideal state of the stomatognathic system.

Theoretically, a purely rotary motion can occur inany position of the mandible heads. A necessary condi-tion for such a motion is a stable constant position ofaxis about which the mandible rotates. The first stage oflimiting motion during posterior opening is performeduntil the dislocation angle of the mandible attains thevalue of ca. 10°. Further abduction is connected with thesecond stage (figure 2, curve 2). At this stage of motionthe rotary axis is dislocated in an anterio-inferior direc-tion. The abduction of the mandible during the secondstage of motion is performed until the widest opening isreached. The limiting motion during anterior openingrepresented by curve 3 (figure 2) is clearly determinedby two terminal positions representing maximum pro-traction and opening. Theoretically, when the mandibleheads are not subjected to linear dislocations and arestable during the occlusion of dental arches, the transpo-sition from the maximum opening to the position ofmaximum protraction of the mandible would be per-formed as a purely rotary motion. However, the maxi-mum protraction of the mandible is affected to someextent by the stylomandibular ligaments which, duringocclusion of the dental arches, retract the mandibleheads. Accordingly, the limiting motion during anterioropening fails to be performed as purely rotary motion.Thus, it may be inferred that the mandible heads attainthe most anterior position during maximum opening, butnot during its maximum protraction. The limiting motiondue to mandible protraction with occlusal contacts (fig-ure 2, curve 4) is determined by the following factors[13]:

• Range of the difference between the posteriorocclusal position and the maximum protraction of themandible (maximum mandible intercuspation).

• Geometry of the cusps of posterior molar teeth.• Occlusion of the incisors of the mandible and

maxilla in the sagittal and horizontal planes.• Geometry of the palatal surface of the anterior

maxillary teeth.• State of dentition within dental arches.The protraction of the mandible in the sagittal

plane is a result of the contraction of the lateral ptery-goid muscles. During the anterior protraction of themandible a slight drop in the recorded trajectory isobserved. The drop results from the alignment of themandibular incisors with respect to the maxillary inci-sors (figure 3). The posterior filaments of the temporalmuscles assist in the retraction of mandible.

J. MARGIELEWICZ et al.12

Fig. 3. Limiting motion during mandible protractionwith occlusal contacts

Motion connected with the protraction of mandible isaffected by numerous factors, and each modification ofany of them results in the trajectory disturbance. In theposterior occlusal position, which is the starting point,the contacts between the posterior molar teeth are ob-served. The first stage of mandible protraction is per-formed until the mandibular incisors come into contactwith maxillary incisors (figure 3a). At the second stageof the motion (figure 3b), the mandibular incisors aredislocated over the incisors of the maxillary bone. Sucha motion is inferiorly oriented and persists until the inci-sors of the mandible and maxillary come into contactwith the incisal margins. The contact between the incisalmargins is the initial point of the third stage of motion(figure 3c), which is characterised by horizontal motion.This stage persists as long as the incisal margins of themandibular teeth attain the position in front of the mar-gins of the maxillary incisors. Starting from this momentthe mandible begins gradually to move upwards. Thismotion persists until the occlusion of the premolar teethis attained. At the last stage of this motion (figure 3e) theshape of the recorded trajectory is considerably influ-enced by the superficial geometry of the premolar teeth.This stage of motion ends when the mandible headsattain the intercuspal position.

3. Mandibular modelin the sagittal plane

In this part of the study, a kinematical analysis ofthe mandible during its abduction and elevation wascarried out. The mandibular motion was modelled by

an open kinematical chain with a variable configura-tion at different time intervals [11], [12]. To formulatea mathematical model the following simplifying as-sumptions were accepted:

• Mandibular motions were analysed in the sagittalplane.

• The mandible is treated as a perfectly rigid solid.The point used to define the beginning of a global

reference system XY (figure 4) with respect to whichthe motion is being described can be freely accepted.According to the authors of this paper, the best loca-tion for the “fixing” of the global coordinates’ systemis the point having position near the central points ofthe mandible heads. Such a place is concurrently thestarting position of the mandible against which thetrajectory is defined. A characteristic feature of sucha location of the global reference system is the factthat a direct access to the initial conditions is obtained.At such a location the initial conditions for linear andangular dislocations of the motion are of zero-value.The initial condition for angular dislocation of themandible is derived directly from its geometry. Theselection of a different point with which the globalreference system will be connected results in the needfor the identification of all initial conditions.

Fig. 4. Kinematical chain modelling mandible motion

A kinematical model of the mandible in the sagit-tal plane, formulated for numerical study, is con-structed of two rotary kinematical pairs and one pro-gressive pair that are responsible for the lineardislocations of the mandible heads. The rotary kine-matical pair (the point A, figure 4), whose position isexplicitly defined by the configuration coordinate ϕ2,is responsible for the abduction and elevation of themandible. The modelling of the elevation motion isdetermined by the configuration coordinates ϕ1 and q.It should be emphasized that at this point the kine-matical chain demonstrated in figure 4 and modellingmandible motion in the sagittal plane can be inter-

Kinematical analysis of mandibular motion in a sagittal plane 13

preted in two ways. In the first interpretation, themodel represents the actual position of mandibularincisors in the sagittal plane (the point B in figure 4).Then, the point A, representing the motion of mandi-ble head, is the result of the projection onto the planeon which the motion is analysed. In the second inter-pretation, the situation is inverse, i.e., the point A rep-resents the actual motion of the mandible head,whereas the point B is the result of projection. Thefact that the numerical models represent the ideal me-chanical states of the mandible should, however, betaken into consideration. A conscious idealization ofmotion in temporomandibular joints incorporated insuch a way is based on the assumption that the ge-ometry of both joints is the same. The application ofmatrix notation in the analysis of mandibular kine-matics allows the definition of the location of any ofits points, with an explicit definition of its orientationwith respect to a static reference system. By means ofthe superposition of elementary relocations of thelocal reference systems, the following dependence isderived:

++−

=

11000100

00

1211212

1211212

lsqscslcqcsc

T0B , (1)

where:q – the length of the segment OA,l – the length of the segment AB,c12 = c1c2 – s1s2, s12 = c1s2 + c2s1.In dependence (1), the first three rows and col-

umns represent an orientation matrix, and the fourthcolumn is interpreted as a location vector. A mathe-matical model of mandible motion in the sagittal planeformulated in such a way is a ground for research oninverse kinematics. Generalized coordinates deter-mining the configuration of the mandibular model forquasi-static positions result from kinematical taskformulated in such a way. One should not overlookthe fact that solving an inverse task they have only thedata for the structure of the model, geometrical pa-rameters and trajectory coordinates. From the mathe-matical point of view the solution of an inverse kine-matical task entails specific difficulties which makethe explicit configuration of a kinematical chain im-possible. Such ambiguities always occur in the case ofredundant systems, i.e., systems with more degrees offreedom than necessary to attain a required position.Practically, this means that the system can realizea given trajectory for numerous configurations. Thepossibility of numerous solutions can also pose the

problems lying in the choice of an optimal configura-tion. The number of configuration solutions depend onthe number of kinematical pairs in the mechanismwhich models a given motion and, additionally, is thefunction of geometric parameters. Theoretically, thereare three situations to be considered which may occurduring the solution of an inverse kinematical task [4]:

• The dimension of a one-column matrix of con-figuration coordinates is smaller than the magnitudeof the location vector. The number of the degrees offreedom in such a system are not sufficient. An in-verse kinematical task can only be solved in specificcases.

• The dimension of a one-column matrix of con-figuration coordinates is equal to the magnitude of thelocation vector. In such a case, the number of the de-grees of freedom correspond with a neighbourhoodvector. The solution of an inverse task is explicitlypossible.

• The dimension of a one-column matrix of con-figuration coordinates is greater than the magnitude ofthe location vector. The kinematical chain has a highernumber of the degrees of freedom than necessary torealize a given task. The solution of an inverse kine-matical task is an infinite set of configurations.

The solutions obtained from numerical methods arebased on the Jacobian matrix [3], [4]. This matrix (2) isachieved by the differentiation of the location vectorwith respect to configuration coordinates. A Jacobian,also named the Jacobian matrix, is one of the mostimportant quantities applied in the analysis and steer-ing of the mechanisms with an open-structure kine-matical chain. It is applied, among other things, to thedesign and realization of smooth trajectories as well asto the identification of singular configurations.

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

i

i

i

zzz

yyy

xxx

ϕϕϕ

ϕϕϕ

ϕϕϕ

...

...

...

21

21

21

J . (2)

The configuration coordinates of an open kine-matical chain (figure 4) realizing a given trajectory aredetermined based on the following iterative equation:

)()( 11

1 iiiPii PPqJqq −⋅+= +−

+ , (3)

where:qi – the configuration coordinates,Pi – the trajectory coordinates,

1−PJ – the pseudoinverse Jacobian matrix.

J. MARGIELEWICZ et al.14

The pseudoinverse Jacobian matrix is deter-mined when the model of the system studied is redun-dant, i.e., when the Jacobian matrix is not a squarematrix:

11 )( −− ⋅⋅= TTP JJJJ . (4)

Dependence (4) is a specific case of the followingequation, assuming that the weight matrix W attainsthe form of an identity matrix:

1111 )( −−−− ⋅⋅⋅= TTP JWJJWJ . (5)

The weight matrix W in (5) is a diagonal matrixinterpreted as the limits imposed on the motion inparticular kinematical pairs. With an increase inweight coefficient the motion in a kinematical pairis subject to limitations. A graphic interpretation ofthe influence of the weight matrix on the type ofmotion of the kinematical chain is demonstrated infigure 5.

Equation (3) is a general recurrent algorithm ena-bling the numerical solution of an inverse kinematicaltask. In simulated calculations of this type of task, oneshould use stable numerical methods, or adequatelylow frequencies of trajectory sampling, because theerrors occurring during its processing have the fea-tures of systematic errors which, as a consequence,lead to an imprecise representation of a given trajec-tory by the kinematical chain.

4. Numerical experiment

The basic action allowing one to carry out a nu-merical study in inverse kinematics is the processing

of the recorded trajectory into a numerical form. Al-though such an operation does not provide any newinformation about the trajectory, it may, however,result in the loss or deformation of information. Theprocessing of a trajectory into a numerical form isusually defined as discretization. Trajectory discreti-zation consists in the selection of the discrete posi-tions in which the trajectory projections onto par-ticular axes of a global reference system are specified.A discrete series of trajectory values should fulfilla basic criterion of conformity. Such a criterion con-cerns the possibility of reproducing its actual shapebased on discreted samples. If the interval between thediscreted samples is incorrectly established, then a pre-cise reproduction of the trajectory is impossible. Suchinaccuracies most frequently appear as attenuatedoscillations after applying an evolution to the trigo-nometric series. The oscillations mostly occur near thelimiting positions of the mandible. When the period oftrajectory sampling is too long, a phenomenon defined

in the theory of signals as frequency masking oraliasing may occur [2], [10]. It should be noted herethat although the aliasing does not have to occur, sucha phenomenon should always be taken into considera-tion.

When there is no digital record of a trajectory, thetrajectory has to be discreted manually which is a time-consuming process. However, the time required forthe discretization of the trajectory can be reduced byinterpolation (figure 6a). Interpolation consists in thegeneration of a full line [9], [14] which crosses thediscreted values defined as the interpolation nodes.Interpolation allows the prediction of the values be-tween the interpolation nodes. One of the interpolationmethods is the application of the Lagrange interpola-tion polynomial:

Fig. 5. Graphic interpretation of the influence of a weight matrix: w2 > w1

Kinematical analysis of mandibular motion in a sagittal plane 15

∑∏

∏

=

≠=

≠=

⋅−

−

=n

iin

ijj

ji

n

ijj

j

yxx

xx

xL0

0

0

)(

)(

)( . (6)

The curve generated in this way is subjected to fur-ther numerical discretization at a new and higher sam-pling frequency. This allows a trajectory with a highernumber of discrete values to be obtained (figure 6b).

Furthermore, the values of new trajectory are equallydistant from each other. Another way enabling us toobtain an increased number of trajectory coordinates isthe application of approximation methods. For that pur-pose, the individual components of a trajectory are ap-proximated by straight lines or parabolas. Owing to thefact that characteristic points of mandible move onclosed paths, the best method for trajectory approxima-tion is by evolving it into a trigonometric series (7). Inorder to apply efficiently the numerical set of values oftrajectory coordinates, this set is processed in a wayaimed at obtaining modified trajectory components inthe form of an explicit function. The use of explicitfunctions in numerical calculations is more convenientthan calculations carried out on a discreted measurementseries:

⋅+⋅+=

⋅+⋅+=

⋅+⋅+=

∑

∑

∑

=++

=++

=++

n

iii

n

iii

n

iii

tiftieetz

tidticcty

tibtiaatx

111

1

111

1

111

1

,)]sin()cos([2

)(

,)]sin()cos([2

)(

,)]sin()cos([2

)(

(7)

where:ai, bi, ci, di, ei, fi – the Euler–Fourier coefficients,n – the number of measurement points,t – the time or number of a sample.Figures 7 and 8 demonstrate the influence of the

quantity of data for discreted trajectory on the qualityof approximation.

The availability of the continuous functions of tra-jectory coordinates allows for the possibility of defin-ing the values for the configuration of a mandibularmodel (figure 4). The configuration coordinates are

derived from equation (3), where a pseudoinverse Ja-cobian matrix has been determined from dependence(5). The same results are obtained by the application ofdependence (4), assuming that the weight matrix is anidentity matrix. It should be emphasized that the itera-tion algorithm applied in kinematical modelling is effi-cient, which is proved by the precise representation ofa given trajectory (figure 9). A representation error forthe trajectory is demonstrated in figure 10.

The influence of the weight matrix coefficients onthe type of the solution of the kinematical chain isshown in figures 11–13.

The basic criterion that should be assumed in theevaluation of the results obtained is the periodicity of thesolution. This criterion must be fulfilled, because eachposition of the mandible that performs a limiting trajec-tory is repeatable. The selection of weight matrix coef-ficients is crucial for this type of trajectory of the man-dibular head, and the determined configuration coordi-nates are essential for its identification (figure 14).

The results obtained from numerical calculationsof inverse kinematics will be incorporated in futurecomputer simulations as the steering values in a dynamicmandibular model.

a) b)

Fig. 6. Discretization of a mandibular trajectory: a) before the interpolation of the trajectory, b) after interpolation

J. MARGIELEWICZ et al.16

Fig. 7. Approximation of measuring data for an insufficient number of measuring data

Fig. 8. Approximation of measuring data for a sufficient number of measuring data

Fig. 9. Trajectory of the mandibular incisors

Kinematical analysis of mandibular motion in a sagittal plane 17

Fig. 10. Representation error of trajectory

Fig. 11. Results of numerical calculations; configuration coordinate ϕ1

Fig. 12. Results of numerical calculations; configuration coordinate ϕ2

J. MARGIELEWICZ et al.18

5. Summary

This paper presents a mathematical approach tomodelling the kinematics of the limiting mandibularmotions in the sagittal plane. The descriptive partapplied a method enabling a kinematical analysisbased on the coordinates of incisor trajectory as theonly available data. The research carried out by theauthors demonstrates that the application of the in-verse kinematical method proves to be a useful diag-nostic instrument enabling the kinematical estimationof mandibular trajectories. An appropriately identifiedmandibular kinematical model provides a basis for thedesign of a dynamic model which allows the analysisof the loads occurring in the temporomandibular jointas well as the estimation of muscular forces. An ap-

propriate choice of the coefficients of a pesudoinverseJacobian matrix enables the path of the mandibularhead to be adjusted to the actual trajectory. Moreover,it can be inferred that the choice of coefficients ofa weight matrix equal to 1 does not fulfil the criterionof periodicity (figure 12). The numerical study of theinverse kinematics of the mandible failed to prove theexistence of a stage connected with a purely rotarymotion. Such a motion would occur, provided that theq- and ϕ1-coordinates in any given stage of motionattain zero values. A comparison of the numericalresults with clinical ones is the final criterion of thepositive verification of the model used in this study. Itproved to be suitable for application in clinical analy-sis. The model presented in this paper provides a firmbasis for further model studies of mandibular motionin three-dimentional space.

Fig. 13. Results of numerical calculations; configuration coordinate

Fig. 14. Results of numerical calculations; trajectories of the central point of the mandibular head

Kinematical analysis of mandibular motion in a sagittal plane 19

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