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# Accelerometers

Industry Articles October 5, 2020

Accelerometers are one of the sensor types used in most inertial navigation systems. As you can guess from their name, they measure acceleration, not velocity. Depending on how long it is since you had to deal with the physical properties of objects, you may recall that the SI unit of acceleration is m/s² (said: metres per second squared). A value of 1 m/s² means that for each additional second that passes, an object’s velocity will increase by an additional 1 m/s (said: metre per second).

Although an inertial navigation system doesn’t directly measure velocity, by keeping track of how much acceleration there is, and how long it lasts, the INS can easily work out what the velocity is by multiplying the acceleration by time.

For example, if it saw an acceleration of 2.5 m/s² for 5 seconds, and assuming the initial velocity was 0 m/s, then the INS must now have a velocity of 12.5 m/s (2.5 m/s² × 5 s = 12.5 m/s).

Distance can also be calculated. It is found using s = 0.5 × at²

where:

• s is distance
• a is acceleration
• t is time

In this case, assuming the inertial navigation system saw the acceleration on the x-axis, it could work out it had moved forwards 31.25 metres (0.5 × 2.5 m/s² × 5 s² = 31.25 m).

So having three accelerometers is very useful, especially when they’re arranged in a mutually perpendicular way, because they allow the INS to measure acceleration in 3D space and calculate the distance traveled as well as current velocity. However, one thing that often confuses people when they first see the data being output from a three-axis accelerometer, is why an axis pointing down shows an acceleration of -9.81 m/s²? To answer that question we need to look at how accelerometers work, and what they actually measure.

At this point, you might think, ‘hang on! Earlier you said accelerometers measure acceleration’. While it is true accelerometers do measure acceleration, we didn’t want to confuse matters by saying that what accelerometers measure is actually acceleration relative to freefall—and that’s why a vertical accelerometer at rest shows a reading of -9.81 m/s². Don’t worry if that doesn’t make sense yet, the next section explains that.

#### Proper acceleration

You have undoubtedly heard the name Sir Isaac Newton before and recall that he wrote some laws of motion. Newton’s first rule tells us that unless some force acts on an object, it will stay perfectly still, or carry on moving at the same speed. In other words, to get something moving, or to change its speed, we need to apply a force. His second rule describes how an object’s acceleration is related to the force acting on it, and the mass of the object. It can be summed up as force = mass × acceleration (F = ma).

Accelerometers measure acceleration relative to freefall using the principle described in Newton’s second law of motion. That is to say, they measure the relative force acting on a known mass, and use that to calculate the acceleration it must be undergoing. To understand this, let’s start by drawing a simple accelerometer.

From the image, we can see the accelerometer contains a known mass, which is attached to a transducer capable of measuring force. However, do note that the mass is constrained within the casing of the accelerometer and can only move left or right—this defines the accelerometer’s measurement axis.

So how does this work in the real world? The image below shows what happens when we place that accelerometer in a car. The car is shown in four states; static, accelerating, cruising at a constant speed and braking. You can see what happens to the mass inside the accelerometer in each scenario.

Static

While the car is static, the mass remains in its centre position as no force is acting on it (at least not along its measurement axis).

Cruising

While the car is cruising at a constant speed, the transducers detect no force and the accelerometer, therefore, the mass registers no acceleration – as with the static car.

Accelerating

When the car accelerates and brakes the mass moves. While accelerating it moves towards the rear of the sensor.

Braking

Under braking, the mass moves towards the front. The harder the car brakes and accelerates, the further the mass is displaced.

Whenever the mass is displaced the transducers measuring the force, register a value. Because the sensor knows the mass and the force acting on that mass, it can easily calculate the acceleration that must be causing the mass to move.

While that seems logical enough, it doesn’t explain why an accelerometer placed vertically on the floor generates a value of 9.81 m/s² even though the floor clearly isn’t moving. And yet an accelerometer in freefall, which clearly is accelerating as it falls through the sky, shows zero acceleration? The answer to that is shown here.

This accelerometer is sitting on the floor. Gravity is acting on both the casing and the mass inside, but neither are in free fall because the ground is stopping the casing from moving and the constraint of the casing is therefore stopping the mass from moving too (unless it tries to move side to side).

In this case, the floor prevents gravity from pulling the casing of the accelerometer downwards—so the casing is not in free fall. However, the mass suspended in the accelerometer is. It can move because the accelerometer has been turned so its measurement axis matches the plane that gravity acts through.

The amount of force applied to the mass will equal the
acceleration due to gravity, and so the sensor will read -981 m/s². So even though the accelerometer is not accelerating, the forces acting on the mass and the casing are clearly different.

This accelerometer is in free fall. Ignoring drag, the only force acting on both the mass and the casing of the sensor is gravity. So, even though the measurement axis is oriented in a way that should measure gravity’s acceleration, the sensor will read 0 m/s² because both the mass and the casing are in free fall. Therefore, there is no relative difference. Or to look at it another way, both the casing and the mass are accelerating at the same rate, so there is no relative difference to measure.

So to summarise, accelerometers are great at measuring straight-line motion, but they’re no good at rotation—that’s where gyros come in.

##### Read the next section in the ‘What is INS?’ series: Gyros 