Biomedical Engineering Reference
In-Depth Information
F
e
+
F
i
m
m
i
γ
i
=
dL
i
dt
e
+
τ
i
m
= M
F
e
+ M
F
i
m
+
τ
where
γ
i
,
m
i
and L
i
stand for the acceleration of segment S
i
center of mass, its mass
and its angular momentum respectively. Hence knowing the external forces, torques,
the mass and the acceleration of the body segment it is possible to deduce (F
i
m
,
τ
i
m
)
if
there is only one unknown for each equation. It means that there is only one force F
i
m
and only one muscle torque
i
m
. However for a given segment S
i
with two adjacent
τ
F
i
m
is the result of two forces: the forces exerted respectively
by each neighboured segment at contact point.
To solve this problem, there are twomainmethods. The first one consists in solving
the system from the extremities (without contact, such as the hands and the head) to
the ground. For body segments placed at the extremities of the skeleton, there is only
one unknown for F
i
m
and
segments S
i
−
1
and S
i
+
1
,
i
m
(associated with the proximal joint attached to the body
τ
i
m
to express their relation
to the joint attached to segments i and j. The problem can then easily be solved by
inverting the above equation:
i
m
could thus be rewritten F
j
→
m
and
j
→
segment). F
i
m
and
τ
τ
F
j
→
i
F
e
−
=
m
i
γ
i
m
j
→
i
i
dL
i
dt
τ
= M
F
e
+ M
F
i
m
+
τ
e
−
m
When dealing with segment S
j
, we can then reuse these results as known forces
F
j
→
i
j
m
. The method is applied until the feet
so that the GRF could be deduced at contact point with the ground. Comparison
with measured GRF is a common method to estimate the errors due to this process.
The second method, named bottom-up method, consists in starting with the segments
which are in contact with the environment (and which external forces are known) and
finishing with the extremities free of any contact. A mixed method can also be used.
F
i
→
m
and torques
j
→
i
i
→
=−
τ
=−τ
m
m
8.5 Global System and Controllers
A second approach to solve inverse dynamics problems is to model the global system
with the Lagrangian formalism (based on the principle of virtual works) in order to
obtain the motion equations, including the torques at any joints
τ
i
:
dt
∂
d
C
∂
˙
q
i
−
∂
C
Q
i
q
i
−
=
0
w
ith i
=
1
...
n
∂
where C is the difference between the potential and kinetic energies, and q
i
is the ith
state of the system. The applied forces and torques Q
i
are expressed in the generalized
coordinates as generalized forces,
