Measuring Motion

As we just mentioned, the key elements of an inertial measurement unit are sensors that measure acceleration, magnetic field and angular velocity. Now, we’re going to talk about how we measure acceleration.

What I have in front of me here is what’s called a quadrotor. It is a flying robot. It’s got four propellers. And, by controlling the speed of the individual propellers, I can make the machine go up, down, left, right and so on. Really critically important to control a machine like this in the air is a sense of balance knowing its orientation in 3D space. So, it has an inertial measurement unit on board. So, somewhere inside this large stack of electronics, there is a board that looks something like this. This board contains three accelerometers, which allow it to determine its roll and pitch angle in space. It contains three gyroscopes which measure its rotational velocity around three orthogonal axis. And, it contains a three-axis magnetometer, so it can determine its orientation where it’s heading with respect to the Earth’s magnetic field.

Now, this little board is customised for this particular quadrotor. A commercial product that performs exactly the same function is a device like this. It’s got a USB port that plugs in to a computer, but internally, it contains three accelerometers, three gyroscopes, and three magnetometers. Now, this same functionality exists in very compact form inside almost all modern smartphones. And, this enables the smartphone compass application so it can find north, it can determine roll and pitch angles. It’s sometimes used for game control and so on.

Imagine now that I’m standing in a lift or an elevator, and I’ve got a very simple dynamical model for my body. It consists of a mass sitting on a spring. The mass represents the mass of my body situated at a point at my centre mass and it’s sitting on a spring which represents the softness of my lower body and my legs. Now, as the lift starts to move upwards, we feel momentarily heavier, that is the weight pushes down on the spring. And, as the lift comes to a stop, we feel momentarily lighter and this is because of inertial effect. It means that it wants to maintain its current velocity. So, when we start off, it wants to maintain its original 0 velocity and that’s what causes it to lag behind the person moving up in a lift. And, that compresses the spring, it make them feel heavier.

As the lift slows down, the weight wants to keep going at the vertical velocity of the lift and that causes that spring to extend and causes us to feel a little bit lighter. Here is a small modern inertial measurement unit, the type that’s quite commonly used in a variety of robot systems. Inside the little box are a number of sensors including accelerometers. So, let’s look inside the box and see how an accelerometer works. And, it looks just like the example we studied before. It contains a mass and a spring. As the body of the accelerometer accelerates upwards, the weight extends the spring. And, as the accelerometer decelerates, the spring pulls the weight upwards. So, the extension of the spring is linearly related to the acceleration of the body of the accelerometer. This is the fundamental principle underpinning all acceleration sensing devices.

Let’s look inside the box and, once again, we see a spring and a mass. The mass is often referred to as the proof mass. And, in this example, it’s got a mass of m and the extension of the spring is denoted it by the symbol x. Now, I’m going to mark the center of mass of the proof mass and its displacement with respect to the ground, I’m going to denote by xm. And, the displacement of the body of the accelerometer, I’m going to denote by the symbol xb. And, so the acceleration of the body is equal to xb double dot and I can write a relationship between xm, xb and x. The spring force acting on the proof mass, I’m going to call fs, and the gravity force acting on the proof mass is the mass multiplied by gravitational acceleration g. I can write Newton’s second law which relates the acceleration of the mass to the two forces acting on at the spring force upwards and gravity acting downwards.

And, for a spring, we know that the force is proportional to the extension of the spring, and we can imagine that as x increases, I’m extending the spring and it is pulling upwards with a greater and greater force. Now, we can combine these equations and do a little bit of rearrangement and simplification. And now, I’m able to write an expression that relates the extension of the spring, x, to the acceleration of the body, a, and the gravitational acceleration, g. It’s really important to note that it cannot tell the difference between acceleration of the body, a, or gravitational acceleration, g, and this is something we’ll return to in just a moment.

Every accelerometer has a sensitive direction and, typically, that’s marked on the body of the accelerometer package. Now, in this particular example, the accelerometer measures acceleration in the vertical direction, but it is insensitive to acceleration in the horizontal direction or in the direction out of the screen. If I place an accelerometer at rest somewhere on the surface of the Earth, the accelerometer records an acceleration of 1g in the vertical or upward direction and this is a little bit unintuitive. If I drop something, it’s going to accelerate in the downward direction that the accelerometer measures 1g in the vertical or upward direction. And, this is a consequence of Einstein’s Equivalence Principle.

And, these two examples are exactly the same as far as an accelerometer is concerned. The accelerometer is sitting on the surface of planet Earth, or if it is sitting in a rocket which is accelerating upwards at 1g. In both these cases, the accelerometer will register an acceleration of 1g in the vertical direction. Gravitational acceleration varies across the surface of the Earth. The Earth is not a perfect sphere. It’s slightly squashed, so the diameter at the equator is greater than the diameter measured through the poles. Because of this, somebody who is standing at the equator is further away from the center of the Earth. So, therefore, they experience a lower gravitational acceleration. So, we can see quite clearly in this figure here, the gravitational acceleration in the equatorial band is significantly lower than gravitational acceleration in the temperate and Polar Regions of the Earth.

The standard gravity value is taken as 9.80665 meters per second squared, but note the significant variation depending on where you are. The low cost accelerometers that you have in your phone and perhaps in the navigation system in your car are based on micro-electromechanical system technology and the whole device is etched out of silicon. The inertial mass or proof mass is supported by flexible spring legs. In the case of horizontal acceleration, the inertial mass will move sideways. And, the change in position is measured capacititively using these sensors here. It’s really useful to be able to measure acceleration in three orthogonal directions aligned with the x, y and z-axis of a right-handed coordinate frame. Today, you can buy tri-axial accelerometers as a single chip. It contains three MEMS accelerometer sensors, electronics, analog to digital converters, filters and a serial IO capability so that you can very easily connect it to a microcontroller.

You may not be aware that you have accelerometers in your head. In the inner ear assembly in each of our ears, there are a pair of accelerometers. And, the physical principles underlying these biological accelerometers are just the same as those we’ve already been talking about. The proof mass is a bunch of calcium carbonate crystals and they are referred to as otoliths, which means literally ear stones. They are sitting on a gelatinous membrane which serves as the spring. And, embedded in that gelatinous membrane are a bunch of hair cells and they measure the displacement of the otoliths with respect to the inner ear. We have two of these accelerometers in each ear, one is called the utricle and one is called the saccule. And, they are oriented almost orthogonally, so one senses the acceleration in the horizontal direction, and one senses the acceleration in the vertical direction.

As you change the orientation of your head, the acceleration signals from the utricle and the saccule vary and allows your brain to determine something about the orientation of your head with respect to the world coordinate frame.