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When shopping for drones, I see a 3 and 6 axis gyro available, what is the difference between the two? Does one have a advantage when flying over another?

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There is actually no such thing as a 6-axis gyro; the correct term for this is "IMU", which stands for "Inertial Measurement Unit". An IMU is a device that combines several inertial sensors that measure the craft's orientation and position in space.

The 6-axis IMU is a composite sensor, which contains two different types of 3-axis sensors on one chip. Usually it is a 3-axis gyroscope (which measures the rate of rotation of whatever it's strapped onto) and a 3-axis accelerometer (which measures its linear acceleration). A 3-axis gyro, on the other hand, is just the gyro and nothing else, as a standalone chip. There are also standalone 3-axis accelerometers¹, but accelerometers are almost never used in flying vehicles alone, without gyroscopes.

The third type of sensor that an IMU can contain is a magnetometer, which measures the earth's magnetic field² and can be used to tell which way is north. A magnetometer can either be present as part of a 9-axis IMU, of another sensor (e.g. in a GPS unit) or an entirely standalone unit. Do you need the accelerometer and magnetometer? Well, it depends.

  • For rate mode (or acro mode, as it's more commonly known), the aircraft only needs to know its rates of rotation along all three axes and matches them to the desired rates set by the pilot via stick positions. For this, just the gyro is sufficient.
  • However, in attitude, angle or autolevel mode (also sometimes called stabilized mode), in which the pilot defines the desired attitude(orientation) relative to the horizon, which the drone will match, a gyroscope alone is insufficient. The flight controller will need to know not only how fast it is currently spinning, but also which way "down" is relative to itself. Due to a phenomenon called integration drift³, the flight controller can't maintain this knowledge precisely enough with just a gyro. It needs an accelerometer on board to correct the gyroscope, so if you want to use these modes, you need both a gyroscope and an accelerometer.
  • A magnetometer (sometimes referred to as a compass) is required for navigation modes, where besides its attitude relative to the horizon the craft has to also know its heading relative to the geographical north. Note that since navigation modes also require GPS, you could forgo the builtin magnetometer in the flight controller, and get a GPS unit with a magnetometer in it instead. Or you can get both, one in the FC and one in the GPS, which will increase the accuracy of the readings. (actually, you can double the other two sensors for the same effect; some flight controllers are manufactured with a twin IMU for that purpose.)

¹: There are also single- and dual-axis versions of each type of sensor — which only means you'll need several of those to make up a complete IMU. They are rarely used nowadays, as the integrated versions are usually better in every way.

²: Or any other magnetic field, for that matter, which is why you need to keep it away from magnets and steel structures, otherwise you'll get rubbish. There's usually little metal in the air other than what's on your aircraft, though, so magnetometers tend to work quite well in UAVs unless you've got a magnet-latched canopy and put the FC next to it or something. Minimizing interference is also why magnetometers are usually integrated into GPS units and mounted as far away from the rest of the aircraft as possible.

³: Integration drift is an issue that plagues inertial navigation since the beginning of time, and is present everywhere where you can't measure something directly, but you can measure how fast it's changing instead (in other words, you can measure the derivative of that thing). So you integrate it (add up the changes over time) to get the original value. However, every measurement comes with an error, and the errors add up as well, so your total error will grow all the time and at some point it will get bigger than the original thing you were measuring.

Intuitively, integration drift is like walking in a forest. When you enter the forest, you (hopefully) have a sense of which way north (or some other reference direction) is. As you walk, you can maintain a sense of direction for some time ("I was walking towards the north, so it was in front of me, now I turned left, north must be to my right now"). However, since you don't know precisely how much degrees you turned, every turn you make will introduce a small error into your direction estimate, and those errors will add up over time. A couple hours later your sense of direction will have drifted far enough from the original that you might be heading in the opposite direction from where you think you're going. Even if you tried to just walk in a straight line all the time you would probably stray off course eventually, because without external reference you have no way of telling whether you've actually been walking in a straight line (this is known as zero drift).

The only way out of this is to have an alternative source of direction knowledge (e.g. a compass or the combination of the sun's position and time of day, or some other observable directionality in your environment like a distant mountain) which you check regularly and use that knowledge to correct the error in your "internal" sense of direction.

A quadcopter in angle mode has the same problem: to stay upright it needs to know is where "up" is, and a gyroscope can only measure (as an approximation) how much degrees it's been rotated from its previous orientation; it does not know (or care) what that previous orientation was. And even if the FC did know its previous orientation, the estimate might have drifted far off from the original. That's where the accelerometer comes in: its job is to measure the earth's gravity, and therefore the direction of "down", and use that information to figure out the initial orientation when the quadcopter is armed (like knowing where north is when entering the forest), as well as correcting possible drift of the orientation obtained from the gyros in flight.

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A three-axis gyro measures the speed of rotation in three-axis - pitch, roll, and yaw.

A six-axis gyro is a slight misnomer, as it is a three-axis gyro with a three-axis accelerometer, which measures acceleration in three axes (x, y, and z). As it can measure the acceleration from gravity, these sensors can be used to determine which way is down relative to the sensor.

Finally, you may see mention of 9 axis gyros. These are 6 axis gyros which include a three-axis magnetometer that can be used to determine heading.

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