Drones

Compare it with a swing. Push a swing and gravity pulls it back through the middle, so it wobbles about the bottom and settles there. The bottom is special: every push away from it costs height, and gravity collects the debt.

A balanced plank has no special position. Level, five degrees over, thirty degrees over: gravity has no preference between them, so nothing collects the debt. Whatever lean a gust starts, the plank simply keeps it, and the next gust adds to it. Machines like this are called unstable, and they are why this course exists.

There are two of them, crossing in the middle like a plus sign. One plank runs left to right and its two ends decide whether the drone leans left or right. The other runs front to back and decides whether the nose points up or down. A fan sits at each of the four ends, so each fan serves both planks at once.

Balancing both planks at the same time sounds harder than balancing one, and it is. Step 2 counts exactly what four fans can and cannot do, and a third turning effect appears there that no see-saw picture predicts.

Not by pushing the four throttles directly. A human reaction takes about a fifth of a second, and Lab 1 shows what the plank does with a fraction of a second to itself. By the time your hand answered, you would be correcting a lean that has already doubled, and each correction would arrive later than the last.

That is why every drone, including the cheapest toy, carries a chip doing the balancing. The sticks on the pilot's radio do not command the fans. They ask the chip for a lean, and the chip does the thousand small corrections a second that hands cannot.

A newton is roughly the weight of a small apple. To hold anything up, the lift has to match the weight, and weight in newtons is the mass in kilograms multiplied by 9.81. The quadcopter in these labs has a mass of half a kilogram, so it weighs 4.91 newtons. To hover, its four motors have to produce that much between them.

Each motor can manage 3.68 newtons flat out, so the four together can lift 14.7 newtons, about three times what the aircraft weighs. That ratio is why a quadcopter can climb steeply, and it is also why it can throw itself into the ground so quickly.

Because a push only ever comes in pairs. To make the air spin one way, the motor has to be pushed the other way, and the motor is bolted to the frame. Sit on a swivel chair, spin a heavy wheel in your hands. You turn the other way. Nothing is leaking or slipping: that is what turning something means.

On a quadcopter this twist is the only way the machine can yaw at all, since no propeller points sideways. Two propellers turn clockwise and two anticlockwise, and when the two pairs work equally hard the twists cancel out. Speed one pair up and slow the other down and the aircraft turns, with the total lift unchanged.

Throughout this course, positive roll means the right-hand side going down, positive pitch means the nose going up, and positive yaw means the nose swinging to the right. Right and left are from the point of view of the aircraft looking forward, the way they are on a ship.

Which way round you choose does not matter, as long as everything agrees. Step 3 is about what happens when one part of the software disagrees with the airframe about it, which is a real and expensive mistake rather than a hypothetical one.

Because of what Lab 3 showed. The four effects are independent, so a roll command adds the same amount to the two motors on one side as it takes off the two on the other. The total is unchanged, and the lift the aircraft gets is whatever the lift demand asked for, whatever is happening in roll.

That is why the four columns of numbers are at right angles to one another in the same sense that north and east are. Travelling east does not change how far north you are. Rolling does not change how much you are lifting.

A throttle is a fraction between nothing and everything. If the four demands add up to more than everything, the extra is thrown away. If they add up to less than nothing, the motor simply stops rather than running backwards. Control Systems calls this saturation, and it is the same wall in a different machine.

It matters more here than it did with a heater, because clipping one motor changes what the aircraft gets from all four. Ask for a big roll while already climbing hard. The two motors that should come down are already at zero, so the roll you get is smaller than the roll you asked for. The lab prints both numbers so you can watch them come apart.

They do one thing more, and it is about clipping. A good mixer does not let a saturated motor quietly change the roll. It notices that the answer does not fit, and scales the demands down until it does, keeping the lift and giving up some of the angle. Step 10 uses exactly that trick to fly on three motors.

Beyond that the mixer stays this simple even on expensive aircraft. It has to run before every single motor write, and there is nothing to gain from making it complicated. The cleverness lives in what decides the four demands, which is Steps 7 to 9.

A bias is a fixed amount added to every reading. A gyroscope sitting perfectly still should report zero and reports 3 degrees a second instead. Nothing is broken: the silicon was etched a fraction off, and the electronics that turn the bending into a number have their own small offset.

You can measure the bias while the aircraft is on the ground and subtract it. That is what a flight controller is doing during the few seconds it asks you to hold still after power-up. That helps a great deal and it does not solve the problem. The bias changes as the chip warms up, and a flight controller warms up as soon as it starts flying.

It is exactly the same shape of failure and worth lining up side by side. There, wheel counts were added up to find a position, and a wheel two per cent too big put the robot further and further from where its own numbers said. Here, rates are added up to find an angle and a bias of a few degrees a second does the same thing to the angle.

The difference is how much it costs. A robot whose estimated position is half a metre from its true one drives into a wall at walking pace. An aircraft whose estimated attitude reads level while the true one leans thirty degrees is accelerating sideways into something, and it has a few seconds.

There are gyroscopes that drift far less. A navigation-grade one uses a laser or a coil of optical fibre rather than a bending piece of silicon. It can hold an angle to a fraction of a degree over an hour. Those are what let a submarine or an airliner navigate with nothing outside to look at.

They also cost thousands of pounds and weigh more than the aircraft in this course. Every quadcopter you have seen uses a MEMS chip costing about the price of a coffee. The whole of Steps 5 and 6 exists to make that chip good enough. Buying the problem away is a real engineering option and it is almost never the one taken.

Inside the chip is a tiny weight held by silicon springs. Gravity pulls the weight one way, the springs stretch, and the chip measures how far in each of three directions. The pattern of those three numbers is the direction of the pull, in the case's own axes. The case is bolted to the aircraft, so the direction of gravity in the aircraft's axes is exactly the aircraft's tilt.

Two of the three numbers are enough for roll: how much of the pull is coming out through the floor, and how much out through the side. If it is all through the floor the aircraft is level. Halfway between the two means forty-five degrees.

Because the weight and the springs have no way of knowing what moved them. Being tilted pulls the weight sideways relative to the case. Being pushed sideways pulls the weight sideways relative to the case in exactly the same way. There is no third measurement available to separate them.

This is not a shortcoming of cheap chips and no better sensor of the same kind fixes it. It is what measuring acceleration means. The word engineers use for what it really reports is specific force. That is the acceleration of the case minus gravity: the pull the case would have to feel to be doing what it is doing.

Yes, and it is the first thing anybody does. Balance the propellers, mount the flight controller on soft rubber pads so the frame's buzz does not reach it, and low-pass filter the readings in software. All three help, and all three are standard.

None of them is free. Soft mounting lets the board move relative to the aircraft, which is its own small lie. A filter that removes vibration also delays the reading, and Step 9 is about what delay costs in a loop like this. Reducing the noise and coping with the noise are both required, which is a pattern worth expecting rather than being disappointed by.

The code multiplies the old estimate by alpha and the accelerometer's answer by one minus alpha. An alpha of 0.99 leaves one hundredth of the gap to the accelerometer closed on every tick, so after a hundred ticks most of it is closed. A hundred ticks at five hundred a second is a fifth of a second, and that is the crossover time.

The lab computes alpha from the seconds you choose and prints it beside the slider. The reason for working in seconds is that the same alpha means different things at different loop rates. An alpha of 0.99 is a fifth of a second at 500 hertz and two whole seconds at 50. A number that changes meaning when you change something unrelated is a good way to lose an afternoon.

Because the nudge is always towards a quantity that is right on average. The gyroscope bias pushes the estimate away at a steady few degrees a second, and the nudge pulls it back in proportion to how far away it has got. Those two balance at a small offset instead of growing for ever.

The offset is easy to predict: the bias multiplied by the crossover time. Three degrees a second and a crossover of half a second leaves about one and a half degrees of lean. That is the price, it is small, and unlike Step 4's error it does not grow with the length of the flight.

A Kalman filter answers the same question and works out the weight for itself rather than being handed one. It carries a running estimate of how uncertain it currently is, and updates that every tick. Each new measurement is then weighed by how noisy it is known to be, against how uncertain the estimate has become. When both sensors are behaving steadily, its answer settles to almost exactly the fixed weight a complementary filter uses.

What it buys is the ability to change its mind. During a violent manoeuvre it can decide the accelerometer deserves almost no say, and while gliding along it can lean on it heavily. What it costs is that it needs honest numbers for how noisy each sensor is. A filter told the wrong noise figures is confidently wrong, in a way that is hard to spot. Plenty of shipping aircraft fly on the three lines in this lab.

Probably, in driving. A driver deciding to be in the next lane does not think about how many degrees to turn the wheel per second. They decide roughly where they want the car and let a faster, more practised sense of steering handle the moment-to-moment work. Two levels, the slower one giving instructions in the units the faster one understands.

Industrial control is full of them. A tank's level loop asks for a flow rate, and a flow loop underneath it works out the valve position. Cascades are worth reaching for whenever there is a fast, well-measured quantity in the middle of a slow, poorly-measured one.

The illusion of a machine that turns at whatever rate it is asked for. That is the trick, and it is why cascades are worth the extra layer. The outer loop does not have to know about inertia, motor lag, propellers or torque. As far as it is concerned it is steering something whose rate it can set.

That illusion is only as good as the inner loop is quick. When the inner loop starts taking a long time to reach the rate it was asked for, the outer loop's picture of the world is wrong. Lab 14 is where that happens.

It is there on real aircraft and it is small. The inner loop has one, to deal with a steady offset: an unbalanced payload, or a slightly weak motor. It is kept small and limited, for the reason Control Systems gives. A running total that grew while the motors were flat out keeps demanding long after the demand became wrong.

These labs leave the inner loop's integral at zero so that nothing in a chart is being caused by something with a memory. The derivative term is here and doing real work, as the damping in the inner loop. On a real aircraft it is the term that needs the gyroscope filtered before it can be used at all.

Because during the gap the rate is itself growing. Halfway through the gap the aircraft is turning at half the rate it will reach by the end. So the angle picked up is not the final rate multiplied by the whole gap. It comes out as half the acceleration multiplied by the time twice over, which is the same sum as how far a dropped stone falls.

The practical result is that halving the loop rate doubles the rate the loop can be surprised by and quadruples the angle. Loop rate does not buy improvement steadily. It buys it in a hurry at the bottom end and hardly at all at the top.

It is one of the several places lag comes from, and it is the one you choose. Deciding every twenty milliseconds means acting on news that is on average ten milliseconds old. Then holding that decision for the rest of the gap. Both halves are delay, and both are paid for by whoever picked the loop rate.

The other sources are not yours to choose. The motors take time, the sensor filtering takes time, and the electronics that hand the number to the motor take time. Step 9 is about the largest of those.

Because past a certain point it stops buying anything and starts costing. A loop faster than the motors can respond is producing decisions nobody can act on, which is what Step 9 measures. And every doubling of the loop rate halves the time available to do the arithmetic in. So a filter has to be cheaper, or the chip has to be bigger, or something has to be left out.

There is a subtler cost too. A derivative term works on the difference between two readings divided by the time between them. At very short times that division is by a very small number, so sensor noise is amplified. Running faster makes the filtering harder rather than easier. That is why real aircraft settle on a few hundred hertz to a few thousand, not on the highest number the chip can manage.

It is the natural way to describe anything that closes a gap in proportion to how big the gap still is. In the first time constant it covers about 63 per cent of the change, in the second about 63 per cent of what was left, and so on. After three time constants it is within a twentieth of where it was going.

Control Systems used the same idea for a thermometer that had to warm up before it could report. The arithmetic is identical. What is different here is which end of the loop it sits at: there it delayed the measurement, and here it delays the action.

Because of Step 7. The inner loop's job is to make the aircraft look like something that turns at whatever rate it is asked for. How quickly it can do that is set by how quickly the thrust can change. Slow the motors and the inner loop gets slower, while the outer loop keeps asking at exactly the rate it did before.

So nothing was mistuned and the tuning is now wrong. The ceiling Lab 14 measured has come down under a gain that has not moved. That is a general shape worth carrying: a tuning is never a property of the software on its own, it is a claim about the software and the hardware together.

Mostly how heavy the propeller is and how far its weight sits from the middle, because that is what has to be spun up or slowed down. A larger, heavier propeller is slower, and so is one made of tough plastic rather than light carbon. The motor's own strength matters too: a motor with more turning force gets there sooner.

The electronics contribute a little. The controller that drives the motor takes a moment to notice a new number. Older ones only accept a new value a few hundred times a second, and that is a delay on top of the lag. Modern digital protocols cut that part down, which is most of what they are for.

Because the four columns are at right angles to one another, which is what Lab 4 was really about. Being at right angles means no mixture of the other three can add up to the fourth. No amount of walking east and north gets you upwards, and this is the same fact.

So the column belonging to the stopped motor is the one direction that has gone, exactly and not approximately. It is easy to read off: it is the four signs that motor carried. Motor 1 was plus height, plus roll, minus pitch, minus yaw, so that combination is what has been lost.

Because it is the only one of the four an aircraft can be wrong about and survive. Lose height and it falls. Lose roll or pitch and it is on its side within a second, and then it also falls. A spinning aircraft that is holding its height and staying level is unpleasant, hard to film with and hard to steer. It is still flying.

Lab 20 lets you insist on one of the others instead, and the equation answers honestly each time. Asking to keep yaw at zero is the most instructive: it can be done, on three motors, as long as the total lift is nothing.

It does, and it has been flown. Research groups have shown quadcopters that lose one propeller, spin up to several turns a second, and keep station well enough to land safely. It is not standard on hobby aircraft. The failure has to be spotted and switched to within a fraction of a second and a wrong detection turns a healthy aircraft into a spinning one.

The version in this lab is simplified in one way, and it is worth saying which. A real spinning aircraft has to be tilted a little. It does not hold its own axes level, because the spin interacts with the turning of the other two. That effect is real, it is called gyroscopic coupling. It is left out here. What it changes is the detail of the attitude held, not the conclusion about which demand has to go.

Because holding still needs a rate and not an angle. The instruction is turn at zero degrees a second, and the gyroscope measures that directly, quickly and without any adding up at all. Step 4's problem was the total. This asks for no total.

What it cannot do is find level. It will hold whatever attitude the aircraft happened to have when the fault arrived. If that was a lean, it will hold the lean and fly sideways the whole way down. Holding still is a much weaker promise than being level, and it is the strongest promise the aircraft can still keep.

By checking its two sensors against each other. There is one signal available and Lab 22 is it: how far the accelerometer's answer sits from the current estimate. On a healthy aircraft those two disagree by a few degrees, because of vibration. On a broken one they disagree by a great deal and stay that way.

The catch is that a real manoeuvre also makes them disagree, for the reason Step 5 gave: an accelerating aircraft has an accelerometer that is wrong. So a detector has to be deaf to a manoeuvre and awake to a fault, and for the first fraction of a second those look the same.

They carry three of everything and vote. With three attitude sources, one going wrong is outvoted by the other two and can be shut out. That is a much stronger position than having two and knowing only that they disagree. Where weight allows, safety-critical systems buy their way out of this problem rather than reasoning their way out.

Even then the answer to a genuine loss of attitude is a procedure rather than a calculation. It is written down in advance, by people who have thought about what is underneath. That is the same conclusion as this step reaches with three buttons: the arithmetic can tell you what each choice costs. It cannot tell you which cost to accept.

The same aircraft as every other lab. Half a kilogram, arms 125 millimetres long, four motors that manage 3.68 newtons each. The angle gain turns degrees off level into a rate request. The rate gain turns degrees per second of rate error into motor commands. The mixer turns those into four throttles, the motors chase them at whatever time constant you chose, and the height loop holds forty metres.

The estimated angle comes from the complementary filter at whatever crossover you set. It is fed by a gyroscope with a three degree per second bias, and an accelerometer feeling whatever vibration you dialled in. The true angle is drawn as well, which no real aircraft has, and comparing the two is the whole reason a simulation is worth having.

The two gains and the crossover, freely, from a laptop. The loop rate, once, from a menu, and it is usually already set sensibly. The other two are jobs with tools. The motor time constant is a different propeller, and the vibration is balancing the ones you have and softening how the board is mounted.

Which is why the vibration slider is worth spending time on. It is the one number a careful builder can improve for nothing but patience, and it moves the best crossover, which moves how well everything else works.

Several things, and it is worth knowing which so a real aircraft does not surprise you. Thrust here is proportional to the command, while a real propeller pushes roughly as the square of its speed. So a real mixer is less even-handed at low throttle than at high. There is no wind beyond a single gust, and real air is never still. The spin in Step 10 does not interact with the other two axes, as noted there. And the arms are rigid, while a real frame flexes and its buzz can find its way into the gyroscope readings.

The largest omission is that this aircraft has no idea where it is. Holding an angle and holding a position are different problems, and the second needs something looking outside: satellites, a camera pointing down, or a rangefinder. That is where this subject joins the Autonomous Systems course. Localisation must represent uncertainty as well as the best position estimate, because the controller needs to know when the estimate is no longer safe to use.

A vision-language or learned control model can interpret a mission, identify a landing area, or propose the next waypoint. The fast attitude and rate loops in this course should remain deterministic. They run more often, enforce physical limits and keep the aircraft stable while a slower model reasons about the mission.

New policies are first tested in simulation and replay, then in shadow mode where their proposals are logged but not executed. A safety monitor checks every accepted command and can return control to a known hover or landing procedure. The Autonomous Systems course covers this layered design.