APPLICATION DESIGN ISSUES - Hard Real-Time Computing Systems: Predictable Scheduling Algorithms and Applications - page 406

robot end-effector

v

object surface

F(t-1) F(t) F(t+1)

Figure 11.8 Force on the robot tool during deburring.

been read, the contact will be detected at the next period; thus, the robot keeps moving
for a distance L T against the object, exerting an increasing force that depends on the
elastic coefficient of the robot-object interaction.

As the contact is detected, we also have to consider the braking space L B covered by
the tool from the time at which the stop command is delivered to the time at which the
robot is at complete rest. This delay depends on the robot dynamic response and can
be computed as follows. If we approximate the robot dynamic behavior with a transfer
function having a dominant pole f d (as typically done in most cases), then the braking
space can be computed as L B

1
2 πf d

. Hence, the longest distance
that can be covered by the robot after a collision is given by

= vτ d , being τ d =

L = L T + L B = v ( T + τ d ) .
If K is the rigidity coefficient of the contact between the robot end-effector and the
object, then the worst-case value of the horizontal force exerted on the surface is F h =
KL = Kv ( T + τ d ). Since F h has to be maintained below a maximum value F max ,

we must impose that

Kv ( T + τ d ) <F max ,

which means

T< ( F max

Kv −

τ d ) . (11.3)

Note that in order to be feasible, the right side of condition (11.3) must not only be
greater than zero but must also be greater than the system time resolution, fixed by the
system tick Q ; that is,

F max
Kv −

τ d >Q. (11.4)

Equation (11.4) imposes an additional restriction on the application. For example, we
may derive the maximum speed of the robot during the deburring operation as

F max
K ( Q + τ d ) ,

v< (11.5)