APPLICATION DESIGN ISSUES - Hard Real-Time Computing Systems: Predictable Scheduling Algorithms and Applications - page 404

v Δ t Δ b

T

V

0

t

obstacle
appears

obstacle brake

pushed

vehicle
at rest

detected

Figure 11.6 Velocity during brake.

If v is the actual velocity of the vehicle and μ f is the wheel-road friction coefficient,

the braking duration Δ b is given by

v
μ f g ,

Δ b =

where g is the acceleration of gravity ( g =9 . 8 m/s 2 ). Thus, the resulting braking

space x b is

= v 2

2 μ f g .
Hence, the total length L needed for a complete stop is

x b

L = v ( T +Δ t )+ x b .

By imposing D>L , we obtain the relation that must be satisfied among the variables
to avoid a collision:

v 2
2 μ f g + v ( T +Δ t ) .

D> (11.1)

If we assume that obstacles are fixed and are always detected at a distance D from the
vehicle, Equation (11.1) allows determining the maximum value that can be assigned
to period T :

T< D

v
2 μ f g −

v −

Δ t . (11.2)

For example, if D = 100 m, μ f = 0.5, Δ t = 250 ms, and v max = 30 m/s (about 108
km/h), then the resulting sampling period T must be less than 22 ms.