Page 181 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 167
mechanism (Figure 3.17).
df( )
̇ x = ⋅ ̇ (3.214)
d
= J ̇ (3.215)

where the J matrix is called the Jacobian of the mechanism. Each element of the Jacobian
matrix is defined as
f ( , , , , , )
6
3
4
5
2
1
i
J = (3.216)
ij
j
where i = 1, 2, ..., m and j = 1, 2, .., n, where n is the number of joint variables. For a six
degrees of freedom mechanism, n = m = 6. If the mechanism has less than six degrees of
freedom, the Jacobian matrix is not a square matrix. Likewise, the inverse of the Jacobian,
J −1 relates the changes in the tool position to the changes in the axis displacements,
̇
= J −1 ⋅ ̇ x (3.217)
If the Jacobian is not invertible in certain positions, these positions are called the geometric
singularities of the mechanism. It means that at these locations, there are some directions
that the tool cannot move no matter what the change is in the joint variables. In other
words, no joint axis variable combination can generate a motion in certain directions at a
singularity point. A robotic manipulator may have many singularity points in its workspace.
The geometric singularity is directly a function of the mechanical configuration of the
manipulator. There are two groups of singularities,
1. Workspace boundary: a given manipulator has a finite span in three-dimensional
space. The locations that the manipulator can reach are called the workspace.Atthe
boundary of workspace, the manipulator tip cannot move out, because it has reached
its limits of reach. Hence, all points in the workspace boundary are singularity points
since at these points there are directions along which the manipulator tip cannot
move.
2. Workspace interior points: these singularity points are inside the workspace of the
manipulator. Such singularity points depend on the manipulator geometry and gen-
erally occur when two or more joints line up.
The same Jacobian matrix also describes the relationship between the torques/forces
at the controlled axes and the force/torque experienced at the tool. Let the tool force be
Force and the corresponding tool position differential displacement be x. The differential
work done is
T
Work = x ⋅ Force (3.218)
Note that the Jacobian relationship
x = J ⋅ (3.219)

and the equivalent work done by the corresponding torques at the controlled axes can be
expressed as
T
Work = ⋅ Torque (3.220)
T
= x ⋅ Force (3.221)
T
= (J ⋅ ) ⋅ Force (3.222)