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13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123

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A new method to solve kinematic consistency problem based on optimizationtechniques and Euler parameters

J. Ojeda* J. Mayo† J. Martínez-Reina‡

University of Seville University of Seville University of Seville

Seville, Spain Seville, Spain Seville, Spain

Abstract — The estimation of the skeletal motion

obtained from marker-based motion capture systems

affects the results of the kinematic and dynamic analysis

of biomechanical systems. A new method based on

optimization techniques and Euler parameters has beendeveloped in this work and compared with other

approaches found in the literature. Results show that for

a theoretical motion this new method gives better results

with a lower level of error. Keywords: Biomechanics, Kinematic consistency, Euler parameters

I Introduction

A typical problem in Biomechanics is the determination

of loads supported by joints or muscle forces necessary to

perform a particular movement. However, directmeasurement of these forces requires invasive techniques

whose implementation is difficult and rarely viable. To

circumvent this problem, inverse dynamics is used toestimate the driving forces required to perform the

movement [1,2].

Since deformation of bones may be neglected for mosthuman motions, the skeleton can be modeled by the

multibody system method. The system consists of rigid

bodies connected by ideal frictionless joints. An array ofat least three markers per segment is needed for the

definition of the position and orientation of a rigid bodyin space. With the mechanical model of the body and the

information of the kinematics, obtained from the markers

and reaction forces on the ground from the force

platforms, the dynamics of the skeletal system, described by its equations of motions, can be inverted to determine

the driving forces in the joints. In this process it isessential to start up with reliable measurements in order

to get accurate results. The estimation of the skeletal

motion obtained from marker-based motion capturesystems is known to be affected by significant errors.

One source of errors is the noise introduced by the

motion capture system. This noise 1is amplificated

because of its high frequency content when the raw

* [email protected]

† [email protected]

‡ [email protected] 13th World Congress in Mechanism and Machine Science,

Guanajuato, México, 19-25 June, 2011

displacement signals are differentiated to obtain thevelocities and accelerations required to solve the inverse

dynamic problem. This high-frequency noise is usually

reduced using different filtering techniques. But there is

another kind of errors which cannot be removed byfiltering because such perturbations typically contain the

same frequencies as those of the movement [3]. These

errors are caused by the skin motion artifact [4,5,6]; due

to skin movements, the markers displace and rotate as arigid body relative to the underlying bone. Due to the

skin motion artifact, the processed kinematic data do not

ensure that the kinematic constraints associated to the biomechanical model are fulfilled, i.e., the kinematic data

are inconsistent with the biomechanical model. The

inverse dynamic analysis also requires that velocities andaccelerations are known. An usual method to obtain those

variables involves the use a polynomial interpolation of

the coordinates and its time derivatives. This procedure

does not ensure that the constraint velocity andacceleration equations are fulfilled, even if the position

data are kinematically consistent. Consequently,

spurious joints reaction forces and net moments-of-force,

associated to the constraint violations, are generated inthe solution of the inverse dynamic problem [7]. These

errors can also have significant effects on the estimation

of the lines of action and lever arms of the muscles and

the forces transmitted in muscles and other structures [8].Efforts have been made to improve measurement

techniques to minimise skin movement artifacts [9] but

they cannot be eliminated unless markers are applied tothe bones directly or through bone-pins [10]. Therefore,

spatial reconstruction of the musculoskeletal system andcalculation of its kinematics and kinetics via a skin

marker based multi-link model should take account of

skin movement artifacts [8].In this work, different methods for generating

kinematically consistent data are compared. The

proposed methods use the filtered data. This is the usualscheme in which filtering and kinematic consistency

numerical schemes are consecutive steps. However, some

authors [11] propose the application of an integratedsmoothing-differentiation-projection approach. They

smooth and differentiate the kinematic signals in a single

step using the Newmark integration scheme and then

project the positions and the obtained smoothedvelocities and accelerations to the biomechanical model

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constraint manifold. Another important characteristic of

the methods used in this work is the model parameters.

The proposed methods adjust the model coordinates to fit

a kinematic model to experimental movement data, but

they consider classical kinematic joints with fixedcenters. Lengths of segments are calculated as the

average of the lengths measured during the capture, or by

the simple direct measurement between anatomical points. Some authors adjust not only the model

coordinates but also joint parameters. Reinbolt et al. [12]

propose a two-level optimization approach thatsimultaneously optimizes joint parameters and motion.

II. Methods to produce kinematic consistent data

using Cartesian coordinates

Different methods can be found in the literature. Theycan be divided into: 1) procedures that minimize the

kinematic inconsistency and 2) procedures that verify

kinematic constraints equations. One method of each

category plus a third one which tries to solve the

limitation of the previous methods will be explained inthis section

A. Adaptation model

Silva and Ambrósio [7] proposed a simplemethodology to systematically ensure the kinematic

data consistency: the kinematic positions are

modified in order to fulfill the constraint equations.Furthermore, the velocity and acceleration of the

system are obtained by using the velocity and

acceleration equations, respectively. The model

coordinates q can be divided into dependent, qd, and

independent, qi, coordinates.

⎢⎢⎣

⎡⎥⎦

⎤=

i

d

q

qq (1)

These authors estimate the independent coordinates

from the measured markers motion. They use thesenon-consistent independent coordinates to calculate

the dependent coordinates solving the nonlinear

position problem.

( ) 0=Φ q (2)

where ф is the system of nonlinear equationsincluding just kinematic constraints. Consistent

velocities q& and accelerations q&& are obtained by

solving the velocity and acceleration equations of the

multibody system:

qq

q

qq

q

&&&&

&

Φ−=Φ

=Φ 0 (3)

This procedure produces consistent data. However,

the biomechanical model is driven by motionhistories calculated from the inconsistent input data,

which are not the true driven motion histories.

B. Local optimization method

The local optimization method, also known as the

segmental optimization method (SOM), looks at the

problem under a different angle. It does not produce

consistent data, but minimizes the inconsistency inall the coordinates of the system, both dependent and

independent.

The method uses least squares optimization to

minimize the difference between the coordinatesobtained with the model and the experimental

marker trajectories. Poses (position and orientation)

of multi-link models are obtained by sequentiallycalculating the separate pose of each segment,

without considering joint constraints [13].

The coordinates vector qi, for each time step, is

estimated from:

( )( )⎭⎬⎫

⎩⎨⎧

−′=⎭⎬⎫

⎩⎨⎧ ∑∑∑

= ==

ii n

j k

jk i jk

n

j

jmm

1

3

1

2

1

2

minmin qm (4)

where ni is the number of markers associated withsegment i (at least three), m an experimental marker,

and m′ the corresponding model marker.

In this work, cartesian coordinates have been used to

describe the system configuration. Euler parameters

have been selected to describe the segmentsorientation.

⎥⎦

⎤⎢⎣

⎡=

i

Oi

ip

rq (5)

where [ ]T

iiiOi z y xr = is the position vector of center

of gravity of segment i, and [ ]T

iieeee p

3210= the

vector of Euler parameters. Because of the use of

Euler parameters, some constraints must be imposed

to the minimization problem in order to get realistic parameters. Thus, the problem would be raised, for

each time step and for each segment, according toequation:

( ) ( )

[ ]

1

3,011

..

)()(minmin

3

0

2

11

2

=

∈≤≤−

⎭⎬⎫

⎩⎨⎧

−+−+=⎭⎬⎫

⎩⎨⎧

∑

∑∑

=

==

i

i

i

n

j

jOi

i

OMjii

T

jOi

i

OMjii

n

j

j

e

ie

t s

mr r p Amr r p Amii

(6)

where Ai(pi) is the rotation matrix of segment i; andr

iOMj the position vector in local coordinates of marker

j that we assume constant and known from anatomical

data.C. Global optimization method

This method takes the best characteristics of the

previous two and solves their limitations. It does not

use inconsistent motion histories to drive the system,

but it completely ensures that the kinematic data are

consistent. The method is based on the search of anoptimal pose of the assembled multi-link model for

each data frame such that the difference between the

measured and model-determined marker coordinates

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are globally minimized in a least squares sense, for

all the body segments.

The minimization problem can be stated as an

optimization problem subjected to the kinematic

constraints:

( )( )

( ) 0qΦ

qm

=

⎭⎬⎫

⎩⎨⎧

−′=⎭⎬⎫

⎩⎨⎧

∑∑∑= ==

..

minmin1

3

1

2

1

2

t s

mm

n

j k

jk jk

n

j

j

(7)

where n is the total number of markers in the systemand q the coordinates vector of the whole system.

Because of the use of Euler parameters, someconstraints must be added in order to get realistic

parameters:

( ) ( )

( )[ ] [ ]

[ ]nbie

jnbie

t s

mr r p Amr r p Am

j

ij

ij

nb

i

n

j

jOi

i

OMjii

T

jOi

i

OMjii

n

j

j

i

,11

3,0,111

..

)()(minmin

3

0

2

1 11

2

∈=

∈∈≤≤−

=

⎭⎬⎫

⎩⎨⎧ −+−+=

⎭⎬⎫

⎩⎨⎧

∑

∑∑∑

=

= ==

0qΦ (8)

where nb is the number of segments.

III. Numerical experiments

To provide a basis for the comparison of the three

proposed methods, computer experiments were

performed in order to set a reference solution. A non-

perturbed three-dimensional reference movement wasgenerated by solving the kinematic problem with selected

driving coordinates. Starting from this unperturbedreference movement, different perturbed movements

were generated by introducing artificial noise into each

three-dimensional marker coordinate. Since Cappozzo etal. [3] found that skin-fixed markers move in a

continuous rather than random fashion relative to theirunderlying anatomical landmarks, a continuous noise

model was chosen,

[ ] [ ]]2,0[

3,1,1]25,0[

]01.0,0[

)cos(

π ϕ

ω

ϕ ω

∈

∈∈∈

∈

++=+=

ik

ik

ik

ik ik ik ref ik ik

ref ik ik

k ni H

m A

t Amnoisemm

(9)

whereref

ik m is the component k of the position vector of

the marker i estimated by solving the constraint

equations, noiseik is the mathematical noise introduced in

the coordinate k of marker i to simulate the skin

movement andik

m the k-component of the position

vector of the perturbed marker, Aik is the amplitude of the

noise, ωik the frequency, t the simulated time, and ϕik the

phase angle. The parameters Aik , ωik , and ϕik wereselected as random numbers, scaled to represent the

motion artifacts anticipated during motion. Since skin

and soft tissue perturbations as large as 2 cm have been

observed experimentally, each amplitude Aik was scaledto be between 0 and 1 cm (i.e. a 2 cm range). Similarly,

since such perturbations typically contain the same

frequencies as those of the movement, each frequency ωik

was scaled to be between 0 and 25 rad/s [14].

Once the perturbed movements were defined, the three

procedures described above were applied to thesetrajectories, comparing the effectiveness in removing the

mathematical noise.

IV. Mechanical model

The lower extremity of the human body was modeled in

this study as a multibody system composed of rigid solidslinked by ideal joints, a procedure widely used to analyze

normal and pathological human gait based on inverse

dynamics. Fig. 1 shows a schematic drawing of the

model adopted here. The use of rigid solids to modellimbs is reasonable for large movements that do not

contain strong impacts such as it occurs during normalwalking. In this case, the effects of movements of the soft

tissues around the bones are small.

Fig. 1. Scheme of the mechanical model adopted

The joints that connect the segments are modeled as

ideal (frictionless) joints, which is not very realistic.

Articular joints present complex contact surfaces as wellas variations of the position of the instantaneous center of

rotation between adjacent segments. However, for largemovements, such as those involved in walking, joints can

be modeled as rotation pairs with a fixed instantaneous

center of rotation, or variable with small errors. This

study used ideal spherical pairs to model the hip, kneeand ankle joints.

The lower extremity model adopted in this study

consists of three rigid bodies representing the thigh, leg

and foot. The system is described by twenty-one

generalized coordinates, seven for each body, describingthe motion of its center of mass and the rotation of the

local system attached to it. The masses, positions of the

centers of mass, and moments of inertia were obtained

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from the literature [15] and linearly scaled to fit the

subject studied [16]. All anthropometric data are shown

in Table I.

Data Thigh Shank Foot

Length (m) 0.46 0.41 0.1

Proximal CM in Z (m) 0.19 0.18 0.05

Proximal CM in X (m) - - 0.117

TABLE 1. Anthropometric data of a subject takenfrom [15]

V. Results

In order to solve the problem of kinematic inconsistency,the three procedures studied here were implemented on

the described mechanical model. Figs. 2-4 show the

temporal evolution of the coordinates of the centers ofmass obtained with the four procedures by imposing the

movement with the analytical equations and adding the

random error as a way of comparing the effectiveness of

each procedure. Figs. 5-8 show the temporal evolutions

of the Euler parameters calculated with the four procedures in the same way. This effectiveness has been

analyzed in a quantitative way by calculating the RMS

error for each procedure (see Table 2). This RMS has been normalized by the averaged value of each

component obtained with the analytical equations.

Fig. 2. Temporal evolution of the component x of the center ofgravity of the different segments. – Solution with no mathematical

noise. – Adaptation model. – Local optimization. – Global optimization.

Fig. 3. Temporal evolution of the component y of the center ofgravity of the different segments. – Solution with no mathematical

noise. – Adaptation model. – Local optimization. – Global optimization.

Fig. 4. Temporal evolution of the component z of the center of

gravity of the different segments. – Solution with no mathematicalnoise. – Adaptation model. – Local optimization. – Global optimization

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Fig. 5. Temporal evolution of the Euler parameter e0 of the different

segments. – Solution with no mathematical noise. – Adaptation model. – Local optimization. – Global optimization


segments. – Solution with no mathematical noise. – Adaptation model.

– Local optimization. – Global optimization





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Gen. coordinate AM LO GO

xthigh 4.39 4.14 0.57ythigh 0.50 1.18 0.20

zthigh 40.60 9.22 8.8

xshank 6.85 0.63 0.75

yshank 0.33 0.29 0.17

zshank 32.08 4.13 3.79

xfoot 33.61 3.72 4.14

yfoot 0.48 0.39 0.24

zfoot 33.05 1.93 2.43

TABLE 2. RMS error in the three procedures

normalized by the averaged value of each component of

the center of gravity for the different segments calculatedwith analytical equations without mathematical noise.

In light of the results obtained, global optimization

clearly provides the best results not only in qualitative

way but also in quantitative terms. Local optimizationrenders worse results because solving each solid

separately does not take into account what happens in therest of the system and therefore some information is lost.

The method of Ambrosio and Silva provides the worst

results alghouth it is the easiest one to implement and itrequires a lower computational effort due to it does not

solve an optimization problem. This suggests that

optimization techniques may be a better approach to

solve the kinematic consistency problem in terms ofaccuracy of the solution.

VI. Conclusions

The present work implements a new approach to solve

the kinematic consistency problem using optimizationtechniques and Euler parameters. The results obtained in

a theoretical motion show that the method is more

efficient compared with other approaches found in the

literature.As future work, the new method will be applied to

experimental measurements of human walking. Along

with that, velocity and acceleration analyses will be

carried out in order to study this approach with moredeepness.

Acknowledgements

This work was funded by the Ministerio Español de

Educación y Ciencia as a part of project DPI2009-11 792

and by the Junta de Andalucía as a part of Excellence

Project TEP03115.

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