Game Development Reference
In-Depth Information
Parallel singularities: The movement of the effector cannot be controlled in certain
directions, even when all the active articulations are blocked, because of these
singularities. These uncontrolled movements of the effector are accompanied by
internal forces in the body of the robot. These singularities only occur on parallel
arms. In addition, they depend on the number of motorised articulations, which
often lead the designers to over-motorise the parallel structures.
It is imperative that there be no singularities in the work space of the robot for the
system to function properly. Direct and inverse static and kinematic models are used
to verify this. They are written as:
V
=
J
· ˙
q art
(8.3)
q art =
˙
G
·
V
(8.4)
The singularities occur when the direct Jacobian matrix J and/or inverse matrix G are
not of full rank. By testing the rank of these matrices, we can thus verify that there
are no singularities in the work space and if required, adjust the parameters of the
robot to remove them. Other than the singularities, no uncommanded or uncontrolled
movement is possible.
Static dimensioning : The second criterion is force capacity, defined in a given
configuration as the minimum force applicable in all directions. To study it, we use
the direct and inverse static models of the robot that help to calculate the operational
forces exerted by the robot on the environment according to the articular torques and
vice-versa. They are written as:
G T
F
=
·
τ art
(8.5)
J T
τ art =
·
F
(8.6)
Equations 8.5 and 8.6 express the transmissions between the articular and operational
variables. Generally, these relationships are also written at the motor level to take into
account the reduction ratios and couplings between the different axes of the robot.
To get an overall view of the results of the limitations at the articular or motor
level, at the level of the effector and dimension the robot actuators, we use the con-
cept of force dimensioning ellipsoid (Gosselin, 2000). This ellipsoid is defined as the
volume generated in the space of the motor torques by a force and a torque given
in the application work space. If we consider F d as the force capacity stated in the
specifications, this ellipsoid can be defined by equation 8.7:
2
F d
F
=
(8.7)
Taking into account equations 8.5 and 8.6, written again at the motor level, this
equation is written as:
τ mot ·
G T )
F d
( G
·
·
τ mot =
(8.8)