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)

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