Receiving a trajectory from an accelerometer and gyroscope (IMU)

Question

I am well aware of the existence of this issue , but mine will be different. I also know that there can be significant errors with this approach, but I also want to theoretically understand the configuration.

I have some basic questions that I find it difficult to answer for myself. There is a lot of information about accelerometers and gyroscopes, but so far I have not found an explanation from the first principles of some basic properties.

So, I have a plate sensor containing an accelerometer and a gyroscope. There is also a magnetometer that I am missing now.

• The accelerometer gives information for each moment of time t about the time acceleration vector a = (ax, ay, az) in m / s ^ 2 according to a fixed coordinate system to the sensor.
• The gyroscope gives a 3D vector in deg / s, which indicates the time rotation speed of the three axes (Ox, Oy and Oz). From this information it is possible to obtain a rotation matrix, which corresponds to an infinitely small rotation of the coordinate system (in accordance with the previous moment). There is some explanation here how to get a quaternion representing R.

So, we know that infinitesimal motion can be calculated, given that the acceleration is the second derivative of the position.

Imagine that your sensor is attached to your arm or leg. At the first moment, we can consider its point in three-dimensional space as (0,0,0), and the initial coordinate system is also attached at this physical point. So, for the first step we will have

r (1) = 0.5 a (0) dt ^ 2

where r is an infinitesimal motion vector, a (0) is the acceleration vector.

In each of the following steps, we will use the calculations

r (t + 1) = 0.5 a (t) dt ^ 2 + v (t) dt + r (t)

where v (t) is the velocity vector that will be somehow estimated, for example, as r (t) - r (t-1)) / dt.

In addition, after each infinitely small movement, we will have to take into account the data from the gyroscope. We will use the rotation matrix to rotate the vector r (t + 1).

Thus, perhaps with a huge mistake, I get some trajectory in accordance with the original coordinate system.

My queries are:

• Am I right in principle with this algorithm? If not, where am I wrong?
• I would really appreciate some resources with a working example where the first principles are not missing.
• How should I use a Kalman filter to get a better path? How do I transfer all IMU data (accelerometer, gyroscope and magnetometer) to the Kalman filter?

Answer 1

Your conceptual framework is correct, but the equations need some work. Acceleration is measured in the platform frame, which can rotate very quickly, so it is impractical to integrate acceleration into the platform frame and rotate the position change. Rather, accelerations are converted to a relatively slowly rotating frame, and there is integration into a change in speed and a change in position. This is usually a locally-level frame (for example, North-East-Down or Wander Aziumuth) or a frame centered in the center (ECEF or ECI). Gravity and Coriolis forces should be included in acceleration.

Conclusions from the first principles can be found in many links, one of my favorites is “Strapdown Inertial Navigation Technology” from Titterton and Weston. The conclusions of the equations of inertial navigation in locally-level and stationary frames are given in Chapter 3.

As you learned in your question, the initial speed is an unknown integration constant. Without some estimation of the initial velocity, the trajectory resulting from the integration of inertial data can be completely erroneous.