ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

The relation between the time-derivative of these angles and the angular velocity for any
instant can be obtained,
where, i, i, i
T
i E
i i i i
(2)
are the time derivatives of i, i, i,
and E i is a 3×3 matrix that
depends on i, i, i.
Analogically, the relation between the angular acceleration vector
and second derivative of the rotation angles can be obtained. It can be shown that it has the
form,
i Ei i i T
i Bi
(3)
are the second order time derivatives of i, i, i.
Also, i B is a vector
. It can be shown that,
where i, i, i
that depends on i, i, iand
i, i, i
ci sici cisi
i i
Bi
si sisi cic
i ii 0 ci
0
i i
Once the first and second time derivatives of , ,
i i i have been obtained, their value for
the next point can be estimated from direct integration. For the time instant i,
2
1
i1 i iTi, i1 i
Ti,
i1
(5)
2
and similarly for and , where for the first point of the trajectory (i = 1) the local
reference system was assumed to have the same orientation as the global one, hence
1, 1, 1
equal to zero. Starting from this value, their values for all the points in the
trajectory can be estimated. For a given instant, the position of each marker can be
determined in the local reference frame by the relation,
r ( R ) ( r r )
L G T G G
Om , L Gm , GO ,
c
where rab ,
is the position vector from point (a) to point (b), expressed in the (c) reference
frame. (O) represents the origin of the reference frame, and (m) can be any body landmark.
As landmarks are subject to errors due to the camera system (stereo-photogrammetric
device) and soft tissue artifact, their position in this local frame is not constant. The mean
of the values calculated over all the trajectory was assumed to represent the rigid body
L
model ( r ) Om ,
. Finally, the trajectory of these landmarks can be reconstructed in the global
reference frame by,
(4)
(6)