Biomedical Engineering Reference
In-Depth Information
8.2.5 Lagrange's Equations
One of the principal forms of the Lagrange equations for a system with
n
generalized coordinates and
m
constraint equations is:
∂ (∂L/∂
˙
q
i
)
∂L
∂q
i
=
−
Q
i
,
(i
=
1,
...
,
n)
(8.8)
∂t
[
]
=
[0]
(8.9)
As we shall demonstrate, these equations consist of
n
second-order non-
linear differential equations and
m
constraint equations. The set has
n
+
m
unknowns in the form of [
q
]
t
and [
λ
]
t
.
8.2.6 Points and Reference Systems
A moving point
pt
in a given Cartesian reference system (RS) has a maximum
of three degrees of freedom (DOF). If a point is constrained to a certain
motion, its DOF are reduced accordingly. A given RS, in addition to the
translational DOF of its original point, has a maximum of three rotational
DOF. In this context, points can represent the origin of an RS, a segment's
center of mass, a point where an external force is applied, a joint center,
a muscle insertion, and other points of interest. An RS that is conveniently
chosen represents either a segment's local reference system (LRS) or the
global reference system (GRS).
A moving reference frame LRS is defined if its point of origin and its
orientation relative to an already defined GRS are given. On the other hand,
a moving point is defined if its RS is given and its local coordinates are
known. By using the notation of linked lists and the indices
i
,
j
,
k
, a point
pt(i )
movingrelativetoanRS
j
is represented by the list:
pt (i )
=
[
j
,
x
i
,
y
i
,
z
i
]
(8.10)
In a similar way, the moving LRS
(j )
with origin at
pt(k )
and given orientation
(usually relative to the zero reference system GRS) is represented by the list:
LRS
(j )
=
[
k
,
θ
1
j
,
θ
2
j
,
θ
3
j
, 0]
(8.11)
For a two-dimensional (2D) system, Equations (8.10) and (8.11) are reduced
to the forms
pt (i )
=
[
j
,
x
i
,
y
i
]
and
LRS
(j )
=
[
k
,
θ
j
]
Starting at some convenient stationary point
pt(
0
)
in the system (point
zero), the GRS is constructed. It is given the index zero. Other points in the
domain of the GRS can be specified by Equation (8.10), where the index
j
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