![](bg2.png)
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,
(B
´
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
5.
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
3
).
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
i
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
i
: Σ
i−1
→ Σ
i
) can be expressed
as, (B
´
aez and Olgu
´
ın-D
´
ıaz, 2013):
A
i
=
R(λ
Ri
, q
i
(t)) d
i
+ λ
Ti
q
i
(t)
0 1
(1)
icSPORTS 2018 - 6th International Congress on Sport Sciences Research and Technology Support
140