Not so smooth criminals: how to use maths to catch a serial killer

The year is 1888, and the infamous serial killer Jack the Ripper is haunting the streets of Whitechapel. As a detective in Victorian London, your mission is to track down this notorious criminal – but you have a problem. The only information that you have to go on is the map below, which shows the locations of crimes attributed to Jack. Based on this information alone, where on earth should you start looking?

The fact that Jack the Ripper was never caught suggests that the real Victorian detectives didn’t know the answer to this question any more than you do, and modern detectives are faced with the same problem when they are trying to track down serial offenders. Fortunately for us, there is a fascinating way in which we can apply maths to help us to catch these criminals – a technique known as geospatial profiling.

Geospatial profiling is the use of statistics to find patterns in the geographical locations of certain events. If we know the locations of the crimes committed by a serial offender, we can use geospatial profiling to work out their likely base location, or anchor point. This may be their home, place of work, or any other location of importance to them – meaning it’s a good place to start looking for clues!

Perhaps the simplest approach is to find the centre of minimum distance to the crime locations. That is, find the place which gives the overall shortest distance for the criminal to travel to commit their crimes. However, there are a couple of problems with this approach. Firstly, it doesn’t tend to consider criminal psychology and other important factors. For example, it might not be very sensible to assume that a criminal will commit crimes as close to home as they can! In fact, it is often the case that an offender will only commit crimes outside of a buffer zone around their base location. Secondly, this technique will provide us with a single point location, which is highly unlikely to exactly match the true anchor point. We would prefer to end up with a distribution of possible locations which we can use to identify the areas that have the highest probability of containing the anchor point, and are therefore the best places to search.

With this in mind, let’s call the anchor point of the criminal z. Our aim is then to find a probability distribution for z, which takes into account the locations of the crime scenes, so that we can work out where our criminal is most likely to be. In order to do this, we will need two things.

A prior distribution for z. This is just a function which defines our best guess at what z might be, before we have used any of our information about the crime locations. The prior distribution is usually based off data from previous offenders whose location was successfully determined, but it’s usually not hugely important if we’re a bit wrong – this just gives us a place to start.
A probability density function (PDF) for the locations of the crime sites. This is a function which describes how the criminal chooses the crime site, and therefore how the criminal is influenced by z. If we have a number of crimes committed at known locations, then the PDF describes the probability that a criminal with anchor point z commits crimes at these locations. Working out what we should choose for this is a little trickier…

We’ll see why we need these in a minute, but first, how do we choose our PDF? The answer is that it depends on the type of criminal, because different criminals behave in different ways. There are two main categories of offenders – resident offenders and non-resident offenders.

Resident offenders are those who commit crimes near to their anchor point, so their criminal region (the zone in which they commit crimes) and anchor region (a zone around their anchor point where they are often likely to be) largely overlap, as shown in the diagram:

If we think that we may have this type of criminal, then we can use the famous normal distribution for our density function. Because we’re working in two dimensions, it looks like a little hill, with the peak at the anchor point:

Alternatively, if we think the criminal has a buffer zone, meaning that they only commit crimes at least a certain distance from home, then we can adjust our distribution slightly to reflect this. In this case, we use something that looks like a hollowed-out hill, where the most likely region is in a ring around the centre as shown below:

The second type of offenders are non-resident offenders. They commit crimes relatively far from their anchor point, so that their criminal region and anchor region do not overlap, as shown in the diagram:

If we think that we have this type of criminal, then for our PDF we can pick something that looks a little like the normal distribution used above, but shifted away from the centre:

Now, the million-dollar question is which model should we pick? Determining between resident and non-resident offenders in advance is often difficult. Some information can be made deduced from the geography of the region, but often assumptions are made based on the crime itself – for example more complex/clever crimes have a higher likelihood of being committed by non-residents.

Once we’ve decided on our type of offender, selected the prior distribution (1) and the PDF (2), how do we actually use the model to help us to find our criminal? This is where the mathematical magic happens in the form of Bayesian statistics (named after statistician and philosopher Thomas Bayes).

Bayes’ theorem tells us that if we multiply together our prior distribution and our PDF, then we’ll end up with a new probability distribution for the anchor point z, which now takes into account the locations of the crime scenes! We call this the posterior distribution, and it tells us the most likely locations for the criminal’s anchor point given the locations of the crime scenes, and therefore the best places to begin our search.

This fascinating technique is actually used today by police detectives when trying to locate serial offenders. They implement the same steps described above using an extremely sophisticated computer algorithm called Rigel, which has a very high accuracy of correctly locating criminals.

So, what about Jack?

If we apply this geospatial profiling technique to the locations of the crimes attributed to Jack the Ripper, then we can predict that it is most likely that his base location was in a road called Flower and Deane Street. This is marked on the map below, along with the five crime locations used to work it out.

Unfortunately, we’re a little too late to know whether this prediction is accurate, because Flower and Deane street no longer exists, so any evidence is certainly long gone! However, if the detectives in Victorian London had known about geospatial profiling and the mathematics behind catching criminals, then it’s possible that the most infamous serial killer in British history might never have become quite so famous…

Francesca Lovell-Read

December 6, 2018 1

Goldbach’s Conjecture: easy but hard

Often in Mathematics problems that are easy to state turn out to be extremely difficult to solve. Two hundred and seventy-five years ago, Goldbach wrote a letter to the famous Swiss mathematician Leonhard Euler in which he wrote the simple statement:

“Every even integer greater than 2 can be expressed as the sum of two primes.”

Just in case you are not up to speed with your maths (and let’s face it why would you be if you’re not a mathematician), let’s break this statement down. The even integers are the numbers divisible by two: 2, 4, 6, 8, …, 256, … and so on. The prime numbers are the ones that can only be obtained by multiplying one by themselves. For example, 3 and 5 are prime numbers because 3=1×3 and 5=1×5 and they have no other representations as a product of two numbers. However, 6 for instance is not prime because 6=1×6=2×3. In fact, all even integers, greater than 2 that were mentioned above, are not primes because they are all divisible by 2 and therefore can be represented as a product of two numbers in at least two ways: 4=1×4=2×2, 6=1×6=2×3, 8=1×8=2×4 etc.

And so, to Goldbach’s conjecture. It says that all even numbers: 4, 6, 8, 10, … can be written as a sum of two primes. Let’s see a couple of examples:

4=2+2

6=3+3

8=3+5

10=3+7

12=5+7

….

A nice way to represent the conjecture visually is through a “pyramid” and because we all love pretty pictures let’s see how this magic happens.

First, we write all of the prime numbers on two of the sides of a triangle as below: 2, 3, 5, 7 etc. We then draw a line leaving each prime number which is parallel to the opposite side of the triangle (stick with me), and finally at the points of intersection of these lines, we write the sum of the numbers. It sounds more complicated than it is as you’ll see with the following example. In the picture below, take the blue line coming out of the number 7 on the left and the red line coming out from the number 11 on the right. They intersect at 18 because 11+7=18. This means that the even integer 18 can be represented as a sum of the two prime numbers 11 and 7. If you look at the intersections of all of the red and blue lines in the pyramid, you’ll see that we actually get all of the even numbers. In other words, any even integer can be written as the sum of two prime numbers, and we can see what those two numbers are by finding the corresponding intersection on our diagram. This is Goldbach’s Conjecture.

It is not very difficult to show that a small even number greater than 2 is the sum of two prime numbers – either by finding the corresponding point on the picture or by trying all of the possibilities. Let’s take 96. We start by checking the smallest prime number 3. 96=3+93, but 93 is not a prime, because 93=1×93=3×31. We continue with the next prime – 5. 96=5+91, which again doesn’t work because 91=1×91=7×13. Next, we try with 7: 96=7+89. Since 89 is a prime, we have obtained a representation of the number 96 as a sum of two primes.

We were able to quickly check whether 96 satisfies Goldbach’s conjecture because the number is relatively small. It becomes much harder to make these checks for larger numbers. It’s been verified with the use of a computer that the conjecture is true for numbers as big as 4×10¹⁸ and this is why the conjecture is believed to be true, but we do not yet have a formal mathematical proof. And being mathematicians, we cannot say something is true until we can prove it.

There have of course been many efforts over the last 275 years to try to prove the conjecture, most of which followed one of two routes. Either by proving that all even integers can be represented as a sum of some number of primes – as a sum of 6 primes (1995, Ramare) and as a sum of 4 primes (Herald Helfgott) – or by proving that almost all even integers can be written as a sum of 2 primes. But, as of yet, the secret formula required to unlock the proof of Goldbach’s Conjecture remains elusive.

You may be wondering why on earth mathematicians are spending their time and effort to prove this seemingly random result about prime numbers? Is it really that important? Whilst you may have a valid point about the applications of this particular conjecture, the value in proving such a result is not in the statement itself, but rather in the new methods, theories and techniques that will need to be developed to solve the problem. So, in 20, 10 or even 2 years from now when you hear that Goldbach’s conjecture has been proved, you should be happy not because we now know for sure that it’s true, but rather because some incredible new area of mathematics has been developed in the process. And who knows, this new area of maths may even pose a new, even more complicated conjecture that will occupy mathematicians for the next 275 years…

Mariya Delyakova

November 15, 2018 0

Take me to your chalkboard

Is alien maths different from ours? And if it is, will they be able to understand the messages that we are sending into space? My summer intern Joe Double speaks to philosopher Professor Adrian Moore from BBC Radio 4’s ‘a history of the infinite’ to find out…

October 31, 2018 0

Complex Numbers – they don’t have to be complex!

The idea of complex numbers stems from a question that bugged mathematicians for thousands of years: what is the square root of -1? That is, which number do you multiply by itself to get -1?

Such a simple question has blossomed into a vast mathematical theory, for the simple reason that the answer isn’t real! It can’t be 1, as 1 * 1 = 1; it can’t be -1, as -1 * -1 = 1; whichever number you multiply by itself, you can’t get a negative number. Up until the 16th century, almost everyone ignored this issue; perhaps they were afraid of the implications it could bring. But then, gradually, people began to realise that there was a whole new world of mathematics waiting to be discovered if they faced up to the question.

In order to explain this apparent gap in maths, the idea of an ‘imaginary’ number was introduced. The prolific Swiss mathematician Leonhard Euler first used the letter i to represent the square root of -1, and as with most of his ideas, it stuck. Now i isn’t something that you’ll see in everyday life in relation to physical quantities, such as money. If you’re lucky enough to have money in your bank account, then you’ll see a positive number on your bank statement. If, as is the case for most students, you currently owe money to the bank (for example, if you have an overdraft), then your statement will display a negative number. However, because i is an ‘imaginary’ unit, it is neither ‘positive’ nor ‘negative’ in this sense, and so it won’t crop up in these situations.

Helpfully, you can add, subtract, multiply and divide using i in the same way as with any other numbers. By doing so, we expand the idea of imaginary numbers to the idea of complex numbers.

Take two real numbers a and b – these are the type that we’re used to dealing with.

They could be positive, negative, whole numbers, fractions, whatever.

A complex number is then formed by taking the number a + b * i. Let’s call this number z.

We say that a is the real part of z, and b is the imaginary part of z.

Any number that you can make in this way is a complex number.

For example, let a = -3 and b = 2; then -3 + 2*i, which we write as -3 + 2i, is a complex number.

As we saw before, complex numbers don’t actually pop up in ‘real-life’ situations. So why do we care about them? The reason is that complex numbers have some very neat properties that allow them to be used in all sorts of mathematical contexts. So even though you may not see the number i in everyday life, it’s very likely that there are complex numbers involved behind the scenes wherever you look. Let’s have a quick glance at some of these properties.

The key observation is that the square of i is -1, that is, i * i = -1.

We can use this fact to multiply complex numbers together.

Let’s look at a concrete example: multiply 2 + 2i by 4 – 3i.

We use the grid method for multiplying out brackets:

	4	-3i
2	2 * 4 = 8	2 * -3i = -6i
+2i	4 * 2i = 8i	2i * -3i = -6 * i * i = -6 * -1 = 6

Adding the results together, we get (2 + 2i)(4 – 3i) = 8 + 6 – 6i + 8i = 14 + 2i.

Therefore, multiplying two complex numbers has given us another complex number!

This is true in general, and it turns out to be very handy. In fact, Carl Friedrich Gauss proved a very famous result – known as the Fundamental Theorem of Algebra because it’s so important – that effectively tells us that the solutions to all equations can be written as complex numbers. This is extremely useful because we know that we don’t have to go any ‘deeper’ into numbers; once you’ve got your head around complex numbers, you can proudly declare that you’ve mastered them all!

Because of this fundamental theorem, our little friend i pops up all over the place in physics, engineering, computer science, and of course, in all sorts of areas of maths. While it may only be imaginary, its applications can be very real, from air traffic control, to animating characters in films. It plays a really important role in much of theoretical mathematics, which in turn is used in almost every scientific discipline. And to think, all of this stemmed from an innocent-looking question about -1; what were they so scared of?!

Kai Laddiman

October 24, 2018 0

Maths proves that maths isn’t boring

If all the maths you’d ever seen was at school, then you’d be forgiven for thinking numbers were boring things that only a cold calculating robot could truly love. But, there is a mathematical proof that you’d be wrong: Gödel’s incompleteness theorem. It comes from a weird part of maths history which ended with a guy called Kurt Gödel proving that to do maths, you have to take thrill-seeking risks in a way a mindless robot never could, no matter how smart it was.

The weirdness begins with philosophers deciding to have a go at maths. Philosophers love (and envy) maths because they love certainty. No coincidence that Descartes, the guy you have to thank for x-y graphs, was also the genius who proved to himself that he actually existed and wasn’t just a dream (after all, who else would be the one worrying about being a dream?). Maths is great for worriers like him, because there’s no question of who is right and who is wrong – show a mathematician a watertight proof of your claim and they’ll stop arguing with you and go away (disclaimer: this may not to work with maths teachers…).

However, being philosophers, they eventually found a reason to worry again. After all, maths proofs can’t just start from nothing, they need some assumptions. If these are wrong, then the proof is no good. Most of the time, the assumptions will have proofs of their own, but as anyone who has argued with a child will know, eventually the buck has to stop somewhere. (“Why can’t I play Mario?” “Because it’s your bedtime.” “Why is it bedtime?!” “BECAUSE I SAY SO!”) Otherwise, you go on giving explanations forever.

The way this usually works for maths, is mathematicians agree on some excruciatingly obvious facts to leave unproved, called axioms. Think “1+1=2”, but then even more obvious than that (in fact, Bertrand Russell spent hundreds of pages proving that 1+1=2 from these stupidly basic facts!). This means that mathematicians can go about happily proving stuff from their axioms, and stop worrying. Peak boring maths.

But the philosophers still weren’t happy. Mostly, it was because the mathematicians massively screwed up their first go at thinking of obvious ‘facts’. How massively? The ‘facts’ they chose turned out to be nonsense. We know this because they told us things which flat-out contradicted each other. You could use them to ‘prove’ anything you like – and the opposite at the same time. You could ‘prove’ that God exists, and that He doesn’t – and no matter which one of those you think is true, we can all agree that they can’t both be right! In other words, the axioms the mathematicians chose were inconsistent.

Philosophers’ trust in maths was shattered (after all, it was their job to prove ridiculous stuff). Before they could trust another axiom ever again, they wanted some cast-iron proof that they weren’t going to be taken for another ride by the new axioms. But where could this proof start off? If we had to come up with a whole other list of axioms for it, then we’d need a proof for them too… This was all a bit of a headache.

The only way out the mathematicians and philosophers could see was to look for a proof that the new axioms were consistent, using only those new axioms themselves. This turned out to be very, very hard. In fact (and this is where Gödel steps in) it turned out to be impossible.

Cue Gödel’s incompleteness theorem. It says that any axioms that you can think of are either inconsistent – nonsense – or aren’t good enough to answer all of your maths questions. And, sadly, one of those questions has to be whether the axioms are inconsistent. In short, all good axioms are incomplete.

This may sound bad, but it’s really an exciting thing. It means that if you want to do maths, you really do have to take big risks, and be prepared to see your whole house of cards fall down in one big inconsistent pile of nonsense at any time. That takes serious nerve. It also means mathematicians have the best job security on the planet. If you could just write down axioms and get proof after proof out of them, like a production line, then you could easily make a mindless robot or a glorified calculator sit down and do it. But thanks to Gödel’s incompleteness theorem, we know for sure that will never happen. Maths needs a creative touch – a willingness to stick your neck out and try new axioms just to see what will happen – that no robot we can build will ever have.

Joe Double

September 6, 2018 0

Spring into action and get ahead of the competition

Wherever we look in the world, we see competition between different groups or beings. Whether it’s two animals trying to earn the right to a watering hole, people trying to assert their social influence, or simply two sports teams playing against each other, this sort of interaction appears in many different situations. As humans, we have a natural desire to rank things that are in direct competition: which is better? Who would win if they faced each other? How does their rivalry compare to others?

We want to know the answers to these questions because it makes us enjoy the competition more, and we feel that we learn more about it. Imagine being able to correctly predict who would win every football match for the rest of the season, you’d probably feel pretty pleased with yourself… But, apart from the inevitable bragging rights, being able to rank competing entities and predict outcomes is an extremely useful skill in many different areas of research, including sociology, economics and ecology.

Of course, you need a bit of maths if you’re going to rank things reliably; you can’t just trust a hunch! There are many different methods that have been used before for rankings, but a group of scientists at the Santa Fe Institute in the USA have come up with a new way of doing it using springs!

So, the ranking system is… a trampoline?! Not exactly. This ingenious method, called SpringRank, treats each interaction as a physical spring, so the model is a whole system of connected springs. Think of a football league: between each pair of teams there is a spring in each direction, and the force of each spring is determined by how many times they have beaten each other in the past. For example, Manchester United have played Liverpool 200 times, winning 80 matches and losing 65. In our spring system, this means that the spring connecting the two teams is biased towards Manchester United – it requires more force to move closer to Liverpool than it does to move towards Manchester United. With this setup, it turns out that the best ranking of the teams is found when you make the total energy in all of the springs as low as possible.

But why use springs? The bonus is that we’ve been studying springs for hundreds of years and so we know the physics behind how they work, which makes it easy to do the calculations. We can use the positions of the springs to work out the rankings of millions of different teams in just seconds! Not only is the maths simple, but it’s also very effective, especially compared to other methods currently used for ranking. In tests run by the researchers, SpringRank performed much better at ranking competitors, as well as predicting the outcomes of future clashes, than existing methods. The data set covered topics as varied as animal behaviour, faculty hiring and social support networks, demonstrating just how versatile the method can be.

This research is a wonderful example of how different areas of science can be combined to create a tool that can actually be put to use in the real world. When learning the subjects separately at school, it’s hard to imagine that you could take centuries-old ideas from physics, turn them into mathematical models, and stick them into a computer program! But here we are, able to work out who is likely to become friends (and enemies), which animals will make it through the heatwave, and whether it’s worth bragging about your favourite team before the game has even happened. So next time you’re challenged to guess the league winner, reach for SpringRank and jump ahead of the competition!

Kai Laddiman

August 29, 2018 0

Getting tattooed for science…

Listen to me being tattooed whilst attempting to describe the process, and hear from my artist Nat on his experience as a tattooist…. all in the name of science.

You can also watch a short video below of the tattoo being done from the perspective of the artist.

Audio edited by Joe Double.

August 16, 2018 0

Alien maths – we’re counting on it

Are we alone in the universe? The possibility that we aren’t has preoccupied us as a species for much of recent history, and one way or another we need to know. The problem is, there is a lot of space, and only so fast you can move around in it, so popping over to our nearest neighbouring star for a quick look around is off the table. We simply don’t know how to communicate or travel faster than light. Nor have we picked up any signals which are identifiable as any sort of message from little green men.

Therefore, perhaps our best chance of making contact with an alien species is to announce ourselves to the universe. If we send out messages to promising-seeming parts of space in the hope that someone will be there to receive them, we might just get a response.

But supposing our signals reach alien ears (or freaky antenna things or whatever), what hope do we have of them being understood? Sure, we might make signals which are recognised as deliberate (and not mistaken for more literal ‘messages from the stars’), but how will they get anything across to aliens whose language is entirely unknown to us?

Scientists in the ‘70s were asking themselves these very questions, and the most promising approach they came up with to get around this problem was one which used maths. In fact, it used an ingenious trick dating back all the way to the Ancient Greeks. The fruit of their labour, broadcast in 1974, was called the Arecibo message.

So, what is it? First off, the Arecibo designers gave up on the hope of sending a written message the aliens could read. Better to stick with pictures – you have to assume aliens will be pretty low down on the reading tree. But this still leaves a conundrum.

When you’re sending a message to space, you have to send a binary signal – a series of ‘1’s and ‘0’s (aka bits) which you hope will start to mean something when it’s processed on the other end. This is precisely how sending pictures over the internet or between computers works too – your message is turned into bits, beamed to the other computer, and then turned back.

And herein lies the problem; the aliens receiving the binary signal won’t have any idea what they’re supposed to do with the bits or how to piece the message back together to make a picture again. You’ve posted them a Lego set but no instructions, and even though they’ve got the bricks there’s no way they’ll figure out whether it was supposed to be built into a race car or a yellow castle. After all, they might not even know what those are!

The way around this is to make the process for turning the message into a picture as simple as possible, so the aliens will be able to guess it. And the way you turn the bits into a picture really is very simple – just write them out in a 23×73 grid, and colour in any square with a ‘1’ in it. Below is what you get (with added colour-coding – see below for what the different parts mean).

White, top: The numbers 1 to 10, written in binary

Purple, top: The atomic numbers for the elements in DNA

Green: The nucleotides of our DNA

Blue/white, mid: A representation of the double helix of DNA. The middle column also says how may nucleotides are in it.

Red: A representation of a human with the world’s pointiest head, with the average height of a man to the left, and the population to the right.

Yellow: A representation of the solar system and the sizes of the planets, with Earth highlighted

Purple, bottom: A curved parabolic mirror like the one used to send the message, with two purple beams of light being reflected onto the mirror’s focus, and the telescope’s diameter shown in blue at the bottom.

Image credit: Arne Nordmann

But how, you might ask, are the aliens supposed to figure out the 23×73 dimensions of the grid? Here is where Ancient Greek maths comes to save us.

The Arecibo message is 1679 bits long. That sounds random, but it is anything but – 1679 is actually the product of two numbers, 23 and 73. Sound familiar? That’s the dimensions of the picture! It’s precisely the fact that 1679 equals 23 times 73 that lets you write out the 1679 bits in a 23×73 grid.

You might be wondering why we used such weird numbers for the sizing. Couldn’t we have chosen nicer, rounder numbers for the picture, like 50×100 say? No. If we did that, the aliens might make a mistake like writing out the bits in a 5×1000 grid or a 500×10 grid, and this would still work numbers-wise because 50×100 = 5×1000 = 500×10.

The key here is that unlike 50 and 100, 23 and 73 are prime numbers. Primes are numbers which can only be divided by one and themselves, like 3 and 5. And most importantly, any number can be split up into primes in a unique way – for instance, 15 is 3×5, and there is no other way to get 15 by multiplying together prime numbers. Likewise, there is no other way to get 1679 than as 23 times 73. So, it is impossible for the aliens to make a mistake when they have to draw out the grid. The Lego set you posted may have no instructions, but you were careful to include parts which can only go together the right way.

An Ancient Greek called Euclid knew this key fact, that numbers split uniquely into primes, over two thousand years ago. The Arecibo designers are banking on the aliens being at least as good with numbers as he was, to be able to decipher the message. Given these are aliens who are capable of picking up a radio signal from space, it seems like a pretty safe bet that they can manage better than an ancient society which believed women have fewer teeth than men because a . It’s a gamble, and it relies on assumptions that the maths we’re interested in is what all species will be interested in – but then what part of blindly shooting intergalactic friend requests into space in the hope someone we’d want to know finds them wasn’t going to be a gamble?

Joe Double

August 9, 2018 0

This robot is a ‘Cheetah’

Robots are developing at an incredible rate, with their ability to perform real-world tasks improving almost by the minute. Such rapid development doesn’t come without downsides, and there are many people who believe that artificial intelligence (AI) could become too powerful, leading to the possibility of robots taking our jobs, or perhaps even taking over the world! Whilst these fears might not be completely unjustified, let’s instead focus on the positives for the time being and marvel at the astonishing accomplishments being made in the field of robotics.

The Cheetah robot, developed by scientists at MIT, is roughly the same shape and size as a small dog, and has been designed to be able to walk across difficult terrains efficiently and effectively. Such a trait is particularly useful when we need to explore dangerous and hazardous environments that may be unsuitable for humans, such as the Fukushima nuclear power plant that collapsed in Japan in 2011. Like all robots, it uses algorithms to help it to navigate, stabilise itself, and ensure that its movements are natural. The latest version, the Cheetah 3, was unveiled in early July, and I think it’s fair to say that it wouldn’t look too far out of place in the animal kingdom!

[Image courtesy of Sangbae Kim, MIT]

Perhaps the most impressive feature of the Cheetah 3 is that the strangely adorable hunk of metal performs the majority of its navigation without any visual input, meaning that it is effectively blind. The researchers at MIT believe that this is a more robust way to design the robot, since visual data can be noisy and unreliable, whereas an input such as touch is always available. Let’s imagine that you are in a pitch-black room; how would you find your way around? Your eyes are pretty much useless, but you can use your sense of touch to feel around the environment, making sure that you don’t bump into walls or obstacles. It’s also important to step carefully, so that you don’t misjudge where the floor is, or tread too strongly and break through something. The Cheetah 3 takes all of this into account as it gracefully glides across even the roughest terrain.

One of the key ideas that was addressed in the new model is contact detection. This means that the robot is able to work out when to commit to putting pressure on a step, or whether it should swing its leg instead, based on the surface that it is stepping onto. This has a massive impact on its ability to balance when it is walking on rough terrain, or one that is full of different obstacles; it also makes each step quicker and more natural. Going back to our dark room, you are likely to step quite tentatively if you can’t see where you are going as this will allow you to react to whatever surface you come into contact with, and adjust your motion as required. With the latest update, the clever ‘canine’ can make these adjustments by itself in a natural manner.

The Cheetah 3 also contains a new and improved prediction model. This can calculate how much pressure will need to be applied to each leg when it experiences a force, by estimating what will happen in half a second’s time. Returning once again to our pitch-black room, imagine how great it would be if you were able to predict what you’re about to step on and adjust your path accordingly – no more treading on sharp objects or stubbing your toe! The scientists tested the power of the new model by kicking the robot when it was walking on a treadmill. Using its prediction algorithm, the Cheetah 3 was able to quickly calculate the forces it needed to exert in order to correctly balance itself again and keep moving. Whilst I can confirm that no animals were harmed in the making of this robot, whether or not the robot itself felt harm is perhaps a question for another day…

The new and improved Cheetah 3 is certainly one of the more remarkable recent accomplishments in the field of robotics. Its natural movements and quick corrections mean that it excellently mimics animal navigation, and it is easy to see how such a robot would be extremely useful for exploring dangerous terrains. Such incredible progress in the study of robotics is as impressive and exciting as it is scary. While it is extraordinary that we are able to replicate animal movements so closely, it has rightly made many people slightly worried; will robots eventually be able to completely replace us? We can only cross our fingers that these critters have no plans for world domination just yet…

Kai Laddiman

July 26, 2018 0