Self-driving cars (or “autonomous cars” or “robotic cars”) are one of the biggest technological challenges of today. The goal is to create a vehicle that can sense and navigate its environment without a human driver. If done well these cars will be safer and more reliable than human drivers. But there are a range of challenges to overcome before this becomes reality. Companies are pouring billions of dollars into research to create the technology, in what may become a trillion dollar market opportunity. Autonomous vehicles are also used in a range of other industries including mining:
Math Scenario 1: Sensing Distance with Lasers and Radar
Most self-driving cars being developed make heavy use of “range” sensors – sensors attached to the car that detect the precise range to obstacles in the environment around the car (such as other cars, pedestrians, fences and barriers).
One of the critical requirements of these sensors is to detect possible hazards far enough in front of the car that the car can react safely. For example, the car needs to be able to detect a pedestrian on the road far enough in front of the car to stop in time.
In Australia, the maximum highway speed is currently 110 km/hr. We can use the stopping distance formula to work out how far a car needs to stop:
- v_end is the final speed of the car
- v_initial is the initial speed of the car
- g is gravity
- d_braking is the estimated braking distance
- f is the friction coefficient. A reasonable coefficient to use is 0.7.
- G is the road gradient (the slope of the road). Let’s use flat ground, which means G is 0.
Since we’re interested in stopping distance, we can set v_end to zero and re-arrange this formula to find the braking distance d_braking:
Converting the speed from km/hr into m/s:
Speed = 110 km/hr * 1000 m/km / 3600 s/hr
Speed = 30.56 m/s
Plugging all the values into the stopping distance formula:
Note that this stopping distance doesn’t take into account the reaction time of the computer’s onboard systems, or the time for the car to apply the brakes once the software has told it to stop. Let’s factor those in as another second of delay:
Total stopping distance = initial speed * reaction & command time + actual braking distance
Total stopping distance = 30.56 * 1 + 68
Total stopping distance = 98.56 m
That means we need a laser or radar sensor that can see at least 98.56 metres in front of the car. One of the most commonly used sensors are the Velodyne LiDARs, which have ranges of 100 or 120 metres.
Math Scenario 2: Camera Resolution for Car and Pedestrian Recognition
At long range, it can be hard for a car to tell the difference between people and other objects such as say a tree trunk, or a bin. Sophisticated cameras are used to help the car detect and recognise people on the road ahead.
For advanced self-driving cars, being able to understand the expression on a human face might be important. One approximate rule of thumb is that you need an image with at least 64 pixels between the human eyes to do good face identification and recognition. The average distance between the eyes of an adult human is a bit over 60 mm.
Let’s say our car has a 20 megapixel camera with a resolution of 5000 (horizontal) by 4000 (horizontal) pixels, and a horizontal field of view of 10 degrees (it’s a zoom camera, specially made for spotting people’s faces). How far away a human face, front on, can we theoretically detect with the camera? We will use some simplifying geometry assumptions, but should get an answer that is in the ballpark of reality.
We require a horizontal resolution of 64 pixels per 60 mm. We can convert this into a resolution per metre:
Resolution = 64 pixels / 60 mm
Resolution = 64 pixels / 0.06 m
Resolution = 1067 pixels / m
Using the diagram above that shows the camera’s horizontal field of view, the distance d to the person, and the horizontal resolution of 5000 pixels, we can write an expression for the horizontal distance captured by the camera (in metres):
tan(5) = opposite / adjacent
tan(5) = opposite / d
opposite = d * tan(5)
Here the opposite side is one half of the horizontal distance captured by the camera, so:
horizontal width captured = 2 * d * tan(5)
horizontal width captured = 0.1750d
We have 5000 pixels to capture this horizontal width, and we need a minimum of 1067 pixels to cover every metre. That means our maximum allowable horizontal width is:
Max width = 5000 pixels / 1067 pixels/m
Max width = 4.686 m
We can substitute this back into the formula to find the maximum distance at which we can detect and interpret a person’s face:
horizontal width captured = 0.1750d
d = horizontal width captured / 0.1750
d = 26.78 m
So for this high zoom camera, we can detect and interpret people’s faces from more than 26 metres away.
Math Scenario 3: Financial Implications of Self-Driving Cars
If self-driving cars become a widespread reality and work as well as the companies developing them hope they will, then they have the potential to have a massive financial impact.
We can do some ballpark calculations for the financial implications in the United States, the likely largest initial market for them.
There are currently about 30000 people killed per year in car crashes, whether as drivers or pedestrians / cyclists. Now it’s unlikely self-driving cars will be perfect, especially initially. But let’s say that they are twice as good as humans are currently, and reduce the death toll by 50%.
Various organisations have to assign a monetary value to a human life – ignoring the host of ethical and moral concerns about equating life with financial figures. The Department of Transportation says a human life is worth about $9.2 million US dollars . So we can estimate that:
Fatal car crash financial savings = #lives saved * saving per life saved
Fatal car crash financial savings = 15000 * 9200000
Fatal car crash financial savings = $138000000000
That’s $138 billion dollars US.
There are far more injuries related to car crashes per year in the United States – around 2.5 million per year. The “financial cost” of an injury is typically (but not always) estimated as being less than a fatality. Let’s assume that an average car crash injury has financial costs associated with it of $50,000 USD – hospital, rehabilitation, time off work, productivity loss and so on .
Let’s say the self-driving cars reduce injuries by 50% as well:
Injury financial savings = #injuries avoided * savings per injury avoided
Injury financial savings = 62500000000
That’s another 62.5 billion dollars US in savings.
Probably the biggest financial difference might be caused by people being able to do other things when in a car, instead of having to pay attention to the driving. If we value an average person’s time at $5 per hour, we need to know how many hours would be saved.
In 2014 the US population travelled an estimated 2.9 trillion miles on the road . To convert that into a time spent in the car, we need to assume an average speed – let’s say 40 miles per hour:
Time in car = distance / speed
Time in car = 2900000000000 / 40
Time in car = 72500000000
That’s 72.5 billion hours per year.
Productivity saving = time in car * productivity cost
Productivity saving = 72500000000 hrs * $5 / hr
Productivity saving = $362500000000
That’s a productivity saving of $362.5 billion dollars.
Adding it All Together
Adding up the fatal, non-fatal and productivity savings, we get a total saving of (B = billion):
Total saving = fatal crash saving + non-fatal crash saving + productivity saving
Total saving = $138B + $62.5B + $362.5B
Total saving = $563B
Little wonder that so many companies are spending so many billions of dollars trying to be the first to roll out a commercially successful self-driving car!
Math Scenario 4: Organ Donation Shortages
One of the unfortunate possible implications of self-driving cars saving lives is having less organ donations for people who critically need them . One in five organs used in transplants are from victims of car crashes.
In the United States, about 6500 people die every year while waiting for an organ donation, and there are more than a hundred thousand people on the waiting list.
In Australia, approximately 60% of families give consent for organ donation from a loved one who has died to go ahead . Organ donation can only happen when a number of circumstances are met, which means the vast majority of people who die in hospital aren’t in a situation where organ donation would have been possible.
If there’s a 50% reduction in fatalities due to autonomous cars, we can calculate the reduction in donations as a result:
Reduction in donations = percentage of organs sourced from car crashes * percentage of people now not dying in car crashes
Reduction in donations = 0.2 * 0.5
Reduction in donations = 0.1 or 10%
A 10% reduction in available organs would have significant effects. Two of the main options being looked into are:
- Increasing the percentage of the population who have consented to donate their organs (an ongoing effort by government), and
- Development of synthetic organs so donors are no longer replied (a long term challenge)