The mathematics of collision - SOTIF for AEB - Article 1/3

In this series of technical articles , we will look into the SOTIF related aspects for AEB. However before we can start dealing into more details of SOTIF and its applicability of AEB , lets first try to clarify what is more critical in context of SOTIF , collision from behind or collision in front .

A] Rear End Collision due to unwanted braking – worst case analysis

1.???? Ego vehicle (A) speed : 100 KMPH , that is 27.77 m/sec ( typically AEB is disabled beyond 90 KMPH)

2.???? Follower Vehicle (B) Speed : 100 KMPH , that is 27.77 m/sec

3.???? Gap : 1 second between two vehicles , that is 27.77 meters - ( Aggressive driving by follower)

4.???? EGO decelerates at :? 6 m/Sec2? - (Around 0.7g - typical for AEB )

5.???? Follower driver worst case reaction time : 1.15 Seconds ( typical reaction time of an experienced driver is 0.68 seconds)

6.???? Follower vehicle decelerates at : 4 m/ Sec^2 ( slower than ego)

?Newton's Laws of motion help us every where :)

The velocity of the ego vehicle (A) ?at time t is given by

x˙A(t) =?27.77??4t.?? ( v= u +at)

The?position of the back of the ego ?vehicle ?at time t is given by

xA(t) =?27.77t??2t^2 +27.77 ( s = ut + ? at^2 + Gap )

The velocity of the follower vehicle (B) ?at time t is given by

x˙B(t) =?27.77??0t = 27.77 , if t<1.15 seconds

x˙B(t) =?27.77??4t , if t >=1.15 seconds

???

The?position of the back of the follower ?vehicle (B) ?at time t is given by

xB(t) =?27.77t??0t^2 ?= 27.77 * t , if t< 1.15 seconds??

xB(t) =?27.77*(t-1.15) – 2(t-1.15)^2 + 27.77*1.15 , if t>=1.15 Seconds???

?

For collision ,We solve the equation?xA(t) =?xB?(t) to find the time at which the follower vehicle (B) will run into the ego vehicle (A)

?

27.77t??3t^2 +27.77 = 27.77*(t-1.15) – 2(t-1.15)^2 + 27.77*1.15

27.77t – 3t^2 + 27.77? = 27.77t – 27.77*1.15 -2(t-1.15)^2 +27.77*1.15

27.77t – 3t^2 + 27.77? = 27.77t -2(t-1.15)^2

?

– 3t^2 + 27.77? = -2*(t-1.15)^2

– 3t^2 + 27.77?? = -2*(t^2 ?-2.23t+1.3225)

– 3t^2 + 27.77?? = -2t^2? +4.46t-2.645

-t^2 -4.46t + 30.415=0

?The time to ?collision turns out ??to be ?3.71 seconds?

The velocity of collision happens to be ??27.77 –( 4*3.71) ?= 12.93 m /sec = 46 KMPH

?If we calculate the same equation with ego deceleration of 4? m/sec2

?27.77t??2t^2 +27.77 = 27.77*(t-1.15) – 2*(t-1.15)^2 + 27.77*1.15

27.77t – 2t^2 + 27.77? = 27.77t – 27.77*1.15 -2*(t-1.15)^2 +27.77*1.15

27.77t – 2t^2 + 27.77? = 27.77t -2*(t-1.15)^2

– 2t^2 + 27.77? = -2*(t-1.15)^2

– 2t^2 + 27.77?? = -2*(t^2 ?-2.23t+1.3225)

– 2t^2 + 27.77?? = -2t^2? +4.46t-2.645

30.415=4.46t

?t = 6.81? Seconds

?Velocity at collision for follower vehicle drops to as low as?? = 27.77-(4*6.81) = 2 KMPH

?

B] Front? End? Collision due to no ?braking – worst case analysis

1.???? Ego vehicle speed : 100 KMPH , that is 27.77 m/sec ( typically AEB is disabled beyond 90 KMPH)

2.???? Front Vehicle Speed : 60 KMPH , that is 16.66 m/sec

3.???? Ego driver suddenly sees the vehicle : 5 meters ?

4.???? EGO decelerates at :? 8 m/Sec2

5.???? Ego driver worst case reaction time : 1.15 Seconds? ( typical reaction? time of experienced driver is 0.68 seconds)

6.???? Front vehicle decelerates at : 0 m/ Sec^2 ( fortunately !)

?16.66 t??0t2 +25 =? ?27.77t– 4(t)^2

16.66t + 5 = 27.77t-4 t^2

16.66t+5 = 27.77t - 4 t^2? ?

4 t^2 – 11.11t + 25 =0

?

Time to collision : 0.56 Seconds , Velocity of collision = 27.77-(8*0.56) m/sec = 83.15 KMPH

The situation is worse if the lead vehicle decelerates or driver does not notice the lead vehicle at all (pitch dark night, truck with no tail lamp - typical Indian Scenario)

• It is clearly evident from above calculations that fear of a follower vehicle colliding ego from behind (in case of AEB) with restricted deceleration is matter of perception & also may become cause of irritation to driver, which can be easily overcome by driver-controlled override like pressing accelerator pedal.
• On the other hand, front collision with undetected object is indeed a life-threatening situation.