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
configuration 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.
209
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6 Analytical Dynamics of Open Kinematic Chains
The number of
degrees of freedom
is the minimum number of independent coor
dinates necessary to describe the dynamical system uniquely. The
generalized veloc
ities
, denoted by ˙
q
k
(
t
)
(
k
= 1, 2, . . ,
n
), represent the rate of change of the generalized
coordinates with respect to time.
The
state space
is the 2
n
dimensional space spanned by the generalized coordi
nates and generalized velocities.
The constraints are generally dominant as a result of contact between bodies,
and they limit the motion of the bodies upon which they act. A
constraint equation
and a
constraint force
are related with a constraint. The constraint force is the joint
reaction force and the constraint equation represents the kinematics of the contact.
Consider a smooth surface of equation
f
(
x
,
y
,
z
,
t
) =
0
,
(6.1)
where
f
has continuous second derivatives in all its variables. A particle
P
is sub
jected to a constraint of moving on the smooth surface described by Eq. 6.1. The
constraint equation
f (x, y, z, t)
= 0 represents a
configuration constraint
.
The motion of the particle over the surface can be viewed as the motion of an oth
erwise free particle subjected to the constraint of moving on that particular surface.
Hence,
f (x, y, z, t)
= 0 represents a constraint equation.
For a dynamical system with
n
generalized coordinates, a configuration con
straint can be described as
f
(
q
1
,
q
2
,...,
q
n
,
t
) =
0
.
(6.2)
The differential of the constraint
f
, given by Eq. 6.1, in terms of physical coordinates
is
d f
=
∂
f
∂
x
dx
+
∂
f
∂
y
dy
+
∂
f
∂
z
dz
+
∂
f
∂
t
dt
=
0
.
(6.3)
The differential of the constraint
f
, given by Eq. 6.2, in terms of the generalized
coordinates is
d f
=
∂
f
∂
q
1
dq
1
+
∂
f
∂
q
2
dq
2
+
...
+
∂
f
∂
q
n
dq
n
+
∂
f
∂
t
dt
=
0
.
(6.4)
Equations 6.3 and 6.4 are called
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 Spring '09
 D.A
 Acceleration, Moment Of Inertia, Strain, Velocity, Lagrangian mechanics, Open Kinematic Chains

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