Chapter 6
Analytical Dynamics of Open Kinematic Chains
6.1 Generalized Coordinates and Constraints
Consider a system of
N
particles:
{
S
}
=
{
P
1
,
P
2
,...
P
i
...
P
N
}
. The position vector
of the
i
th particle in the Cartesian reference frame is
r
i
=
r
i
(
x
i
,
y
i
,
z
i
) and can be
expressed as
r
i
=
x
i
ı
+
y
i
j
+
z
i
k
,
i
=
1
,
2
,...,
N
.
The system of
N
particles requires
n
=
3
N
physical coordinates to specify its po-
sition. To analyze the motion of the system in many cases, it is more convenient to
use a set of variables different from the physical coordinates. Let us consider a set
of variables
q
1
,
q
2
q
3
N
related to the physical coordinates by
x
1
=
x
1
(
q
1
,
q
2
q
3
N
)
,
y
1
=
y
1
(
q
1
,
q
2
q
3
N
)
,
z
1
=
z
1
(
q
1
,
q
2
q
3
N
)
,
.
.
.
x
3
N
=
x
3
N
(
q
1
,
q
2
q
3
N
)
,
y
3
N
=
y
3
N
(
q
1
,
q
2
q
3
N
)
,
z
3
N
=
z
3
N
(
q
1
,
q
2
q
3
N
)
.
The
generalized coordinates
,
q
1
,
q
2
q
3
N
, are the set of variables that can
completely describe the position of the dynamical system. The
confguration space
is the space extended across the generalized coordinates. If the system of
N
particles
has
m
constraint equations acting on it, the system can be represented uniquely by
p independent generalized coordinates q
k
,(
k
=1
,2,.
..
,
p
), where
p
=
3
N
−
m
=
n
−
m
.
The number
p
is called the number of degrees of freedom of the system.
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