The Collision Detection Problem

• Human reaction time approx. is 1 sec. so a good system should warn us way ahead of that.

A collision avoidance system (CAS) is an active safety feature that warns drivers or even triggers the brake in the event of an imminent collision with an object in the path of driving. If a preceding vehicle is present, the CAS continuously estimates the time-to-collision (TTC). When the TTC falls below a lower threshold, the CAS can then decide to either warn the driver of the imminent danger or - depending on the system - apply the vehicle brakes autonomously. For the engineering task you will be completing in this course this means that you will need to find a way to compute the TTC to the vehicle in front.

Constant velocity vs. constant acceleration

To compute the TTC, we need to make assumptions on the physical behavior of the preceding vehicle. One assumption could be that the relative velocity between the yellow and green vehicle in the above figure were constant. This would lead to the so-called constant velocity model (CVM) which is represented by eq. 1 in the following diagram.

We can thus expand on our CVM by assuming velocity to be a function of time and subtract a second term in eq. 2 which is the product of a constant acceleration and the squared time dt between both measurements. Eq. 3 displays velocity as a function of time, which is also dependent on the constant acceleration we used in eq. 2. This model is referred to as constant acceleration model (CAM) and it is commonly used in commercially available collision detection systems. On a side note, if we were using a radar sensor instead of a Lidar, a direct measurement on velocity could be taken by exploiting a frequency shift in the returning electromagnetic wave due to the Doppler effect. This is a significant advantage over sensors such as Lidar, where velocity can only be computed based on (noisy) distance measurements.

In this course, we will be using a CVM instead of the CAM as it is much simpler to handle with regard to the math involved and with regard to the complexity of the programming task ahead of you. For small instances of dt we will assume that the CVM model is accurate enough and that it will give us a decent estimate of the TTC. Should you be involved in building a commercial version of such a system at a later stage in your career however, keep in mind that you should be using a constant acceleration model instead.