An Alternative Method for Kinematic Modelling Applied to the Human

Joint Position

Bibiana Hern

andez-Hern

andez

1,2

, Cesar Ramos-Villa

1,2

, Dulce Martinez-Peon

1,2

Eduardo Torres-Sarmiento

1,2

and Ernesto Olgu

ın-D

ıaz

Division of Studies and Postgraduate Research, Institute of Technology of Nuevo Leon, Mexico

National Technological Institute of Mexico, Mexico

Robotics and Advanced Manufacturing Group, Cinvestav, Coahuila, Mexico

Keywords:

Kinematic Modelling, GRyMA Methodology, Biomechanics.

Abstract:

The kinematic description of a mechanical structure is an essential part of the motion analysis. In human-

like structures, kinematics works as a biomechanical base for the analysis of the human motion in areas

like rehabilitation and sports. While nowadays the standard is based on the Denavit-Hartember convention,

which has been deﬁned for industrial mechanical use, this approach may need virtual reference frames when

having non orthogonal systems and complex geometries like human body. This work presents a kinematic

analysis focused on lower limbs using an alternative to the Denavit-Hartenberg convention for reference frames

assigment in kinematic modelling, called the GRyMA methodology. Finally, the paper also shows Matlab

-

based simulation of a CAD model emulating the human lower limbs motion. The kinematic analysis could be

used in the assessment of the joint position of individuals with some walk or sports disability, and therefore

also in the correct treatment or posture improvement.

1 INTRODUCTION

Biomechanics is the science that involves the study of

the mechanical features of living organisms, mainly

focused on human anatomy, (Hall, 2012). There are

two important concepts within this discipline: Ki-

nematics, a part of mechanics ﬁeld that studies the

motion of objects regardless of the forces applied to

them, including displacement, velocity and accelera-

tion; and Kinetics, which studies the forces that pro-

duce the body motion and its changes, (Bergmann and

Peterson, 2011).

Human motion analysis observes in detail the hu-

man motion in order to gather quantitative informa-

tion about the mechanics of the body when executing

different tasks. In this branch, the study of the human

gait is very important because of its use in clinical ma-

nagement like planning and treating individuals with

some walking or sports disability; or to estimate the

joint positions of a healthy runner, (Lu and Chang,

2012). The kinematic ﬁeld provides an accurate des-

cription of the human motion and it is fundamental to

understanding its biomechanics, (Knudson, 2007).

In sports, posture analysis is fundamental and

Figure 1: 12 DoF lower limbs model (6 for each leg).

must be obtained precisely. Some analysis techni-

ques includes numerical methods like small series of

kinematic patterns, (Bharatkumart et al., 1994), pre-

Hernández-Hernández, B., Ramos-Villa, C., Martinez-Peon, D., Torres-Sarmiento, E. and Olguín-Díaz, E.

An Alternative Method for Kinematic Modelling Applied to the Human Joint Position.

DOI: 10.5220/0006937301390144

In Proceedings of the 6th International Congress on Sport Sciences Research and Technology Support (icSPORTS 2018), pages 139-144

ISBN: 978-989-758-325-4

139

dictions of length patterns of in-vivo ligaments with

global position vectors, (Lewis and Lew, 1978), mo-

dels estimation for anatomical joints with minimum

squared error, (Sommer and Miller, 1980). Scott,

purpose a software to analyse musculoskeletal struc-

tures from input parameters provided by user where

kinematics is solved using transformations (orthogo-

nal translations and rotations) between axes having

as a reference the HUMERUS and ULNA segments,

(Delp and Loan, 1995). Hanavan takes advantage of

computers as an useful tool when modelling human

body and purposes a 15 geometric solid mathemati-

cal model for body modeling, based in antropometric

dimensions, and describes body positions with Euler

angles, (Hanavan JR, 1965); while Bogert and their

colleagues designed a software system able of calcu-

lating forward and inverse kinematics of 44 degrees of

freedom (DoF), (Bogert et al., 2013). Baldisserri and

Castelli made a 3D kinematic model considering four

bones (tibia, ﬁbula, talus and calcaneus) to emulate

passive motion; the analysis was made using Denavit-

Hartenberg transformation matrices, (Baldisserri and

Castelli, 2010).

In addition to the aforementioned techniques,

there are alternative marker-based methods, (Rab

et al., 2002) that can be subdivided in two catego-

ries: contact based techniques, including accelerome-

ters and goniometers sensors, and non-contact based

techniques using active LEDs (Light-emmiting dio-

des) or passive markers (tags), (Prakash et al., 2015).

Nonetheless, there is still much to do in technologi-

cal improvement, mathematical modelling of muscu-

loskeletal system and more techniques to quantify and

reduce measurement errors, (Lu and Chang, 2012).

In Kinematics, the most common method to des-

cribe the link structure of an articulated body is the

well known Denavit-Hartenberg method, (S.Kajita,

2014); although it is not always practical when ana-

lysing complex systems. Therefore the present work

proposes an alternative method in the assignation of

reference frames known as GRyMA methodology,

aez and Olgu

ın-D

ıaz, 2013).

This work is organized as follows. In Section 2,

the alternative approach of the forward kinematics is

explained, detailing the algorithms to allocate the re-

ference frames as well as the structure of the homo-

geneous tranformations. Section 3 presents a case of

study representing the basic motion of human lower

limbs by appliying the alternative GRyMA methodo-

logy. In Section 4, simulation on a 3D CAD model

is presented using the GRyMA procedure in order to

show the simplicity and functionality of the proposed

method. Finally, the conclusions are stated in Section

2 METHODOLOGY

DESCRIPTION

The most common method used to calculate kinema-

tics within a kinematic chain consisting of articulated

bodies is by using order 4 homogenous transformati-

ons as:

A =



R d

0 1



∈ SE(3)

Each of these transformations represents a rotation

(through the 3 DoF Rotation matrix R ∈ SO(3)) and a

translation (through the 3 DoF displacement d ∈ R

Then by assigning a reference frame (normally under

the right hand rule) in each rigid body in the kine-

matic chain, it arises an homogenous transformation

between these frames upon which any position and

hence its time derivative values can be calculated.

These homogeneous transformations needs 6 li-

neally independent parameters to be described. In

this sense the Denavit-Hartenberg convention impo-

ses two restrictions on each two consecutive frames,

which reduces the complexity of each homogeneous

transformation to only four parameters. Even more,

for articulated bodies in a kinematic chain, only one

of these parameters is variable becoming the relative

joint value, while the remanent 3 constant parame-

ters describes the kinematic chain. This methodology

has been proven to be very useful and simple even in

complex systems. However it has been designed spe-

ciﬁcally for mechanical systems where the two con-

straints needed are always fulﬁlled.

These constraints are seldom fulﬁlled on complex

systems like biomechanical ones, for which most of

the time an additional virtual frame in each articula-

tion has to be included to guarantee the validity of the

methodology. In these cases, the number of parame-

ters needed to describe a single articulation increases

to 8 (with only one variable) which renders the ho-

mogeneous transformation more complicated than the

original of 6 DoF without the DH convention.

An alternative approach for the assignation of

these frames is the GRyMA method (After Grupo de

Rob

otica y Manufactura Avanzada), where the origin

of every reference frame Σ

is placed at any user cho-

sen point along the articulation axis, and always pa-

rallel to the inertial frame at the home position of the

system. Then the homogeneous transformation of the

child/parent frames (A

: Σ

i−1

→ Σ

) can be expressed

as, (B

aez and Olgu

ın-D

ıaz, 2013):



R(λ

, q

(t)) d

+ λ

(t)

0 1



(1)

icSPORTS 2018 - 6th International Congress on Sport Sciences Research and Technology Support

140

(a) using DH convention (b) using GRyMA methodology

Figure 2: Reference frames, note that in ﬁgure b all the axes including global reference frame are oriented in the same

direction.

where q

is the variable generalized coordinate corre-

sponding to the articulation; the 3D distance d

∈ R

in coordinates of the parent frame i-1, is the con-

stant position of the origin of the child’s frame; the

unit director vectors λ

and λ

(composed of 3 ele-

ments each), also in coordinates of parent frame, de-

ﬁne whether the articulation is rotational or prismatic

(respectively) and the rotation matrix is deﬁned upon

the axis/angle representation.

Then the mode and direction of the articulated

joint is given by the kinematic director λ

(θ

) =

(λ

, λ

)

∈ R

, where λ

can be codiﬁed with the

scalar parameter θ

according to the Table 1, using

only 4 constant parameters and one variable for each

articulation. The GRyMA methodology uses Algo-

rithm 1 to place the reference frames.

Table 1: Kinematic director.

θ 0 1 2 3 4 5 6

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

Where:

θ Indicate which axis will have a movement

Indicate rotation movement

Indicate translation movement

Algorithm 1: GRyMA methodology for Frame Assignment,

aez-Golubowski, 2015).

1: Identify the motion axes of the articulations in the

system.

2: Assign the root (not always the inertial one) re-

ference frame Σ

in a way that the position and

orientation are strategically deﬁned with the mo-

tion axes of the system. Allocating it along the

ﬁrst articulation is useful because it would sim-

plify the parameters.

3: Assign each reference frame Σ

of the i-link with

the origin along the articulation axis and taking

the same orientation as deﬁned in the inertial

frame. If possible the origin may be allocated to

intersect the motion axis of the parent frame too,

in order to make one of the three position para-

meters equal to zero and simplify the parametric

deﬁnition.

4: Deﬁne the distance vector d

∈ R

from the Σ

i−1

frame to the Σ

frame.

5: Codify the direction parameter θ according to Ta-

ble 1.

3 EXPERIMENTAL WORK

In this section, GRyMA methodology is applied to the

3D study case model, which stands for 7 segments of

the human lower limbs.

The assignment of the reference frame Σ

deﬁnes

the orientation of the other frames at initial or home

An Alternative Method for Kinematic Modelling Applied to the Human Joint Position

141

position. The restriction is that all articulation axes

have to be aligned to any axis of the frame Σ

in order

to simplify the analysis to 4 constant parameters and

one joint variable, shown in Table 2 according to the

Table 1, for the same case of study. Figure 2(b) shows

the corresponding frame assignment.

Table 2: GRyMA kinematic parameters.

Joint d

θ Σ Parent

1 d

6 Σ

2 d

4 Σ

3 d

5 Σ

4 d

5 Σ

5 d

5 Σ

6 d

4 Σ

7 d

6 Σ

8 d

4 Σ

9 d

5 Σ

10 d

x10

y10

z10

5 Σ

11 d

x11

y11

z11

5 Σ

12 d

x12

y12

z12

4 Σ

The homogeneous transformation matrixes that

deﬁne the position and orientation of the links refe-

rence frames are obtained after equation (1). The ho-

mogeneous transformation matrix of the ﬁnal effector

frame respect the root frame, its obtained again after

expression (2).

T =

∏

i=1

i−1



(q) d

(q)

0 1



(2)

Given this, the value of position d

and orienta-

tion R

of the ﬁnal effectors (also known as Forward

Kinematics) is obtained as functions of the joint coor-

dinates (θ

. . . θ

4 RESULTS

To test the kinematics results obtained by the GRyMA

methodology, a 3D CAD model for the 7-segment

model was made using V-Realm Builder tool of Mat-

lab

 to simulate the motion in human lower limbs.

The joint structure shown in Figure 1 has 12 actuated

DoF (the 6 DoF of the root frame has been constraint

to have no motion), the hip joint has 3 DOF which al-

lows the abduction-adduction, ﬂexion-extension and

rotation of the upper leg segment; the knee joint has

1 DoF for the ﬂexion-extension lower leg segment;

and the ankle is provided with 2 DOF, including the

frontal ﬂexion-extension and rotation of the foot.

The block diagram made in Simulink

 to simu-

late the positions of the links can be appreciated in

Figure 3.

Figure 3: Simulation block diagram.

Anthropomorphic measures were taken into ac-

count to set the lenght of the links of the 3D CAD

model as well as the tridimensional translation d

∈

for the reference frames of the system. Hu-

man motion in lower limbs like rotation, extension-

ﬂexion, adduction-abduction and inversion-eversion

were simulated. The results of the implementation of

the GRyMA methodology are presented in Figure 4,

which displays an aproximation of the desired beha-

vior of the system in the virtual world when perfor-

ming different postures.

5 CONCLUSIONS

The Denavit-Hartenberg convention is the most com-

mon mathematical methodology in the assignment of

frames to produce the kinematic study of articulated

rigid multi-bodies systems but when it comes to more

complex systems, auxiliar reference frames are nee-

ded, which can generate confusion when deﬁning the

centers of mass of the links, besides it makes the kine-

matic analysis harder by elevating the parameters by

4 each time an auxiliar frame is used.

One of the advantages of the GRyMA methodo-

logy is that the allocation of the children frames can

be done (by deﬁnition of the GRyMA methodology)

in such a way that also intersects the motion axis of

the parent frame, this allows to make one of the three

position parameters equal to zero and simplify the pa-

rametric deﬁnition.

Having a simpliﬁed homogeneous transformation

becomes handy for the solution of the inverse kine-

matics, as an alternative to existing solutions.

ACKNOWLEDGEMENTS

We would like to thank CONACYT for the scho-

larships granting with numbers 612807, 605806 and

607608, which supported the studies of the students

during their master.

icSPORTS 2018 - 6th International Congress on Sport Sciences Research and Technology Support

142

(a) (b) (c)

(d) (e) (f)

Figure 4: Representation of different movements in virtual model by applying the GRyMA methodology to the kinematic

modelling: a), d) Sagittal displacement of the knee and the ankle respectively; b), e) Sagittal displacement of the hip; c)

Frontal displacement of the hip; f) Horizontal displacement of the hip and Frontal displacement of the ankle.

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