All About IMUs - Accelerometers and Gyros

As part of a navigation solution, Inertial Measurement Units (IMUs) are a crucial integrated sensor unit that makes dead reckoning possible. At minimum, IMUs integrate accelerometers and gyroscopes, but they often include magnetometers and pressure sensors, offering a robust sensor fusion package for an Inertial Navigation System (INS). To understand how IMUs are used to create navigation solutions, let’s take a look at each of these individual sensors: How they work, their key performance parameters, their susceptibility to error, and how their measurements are used. First up, let’s consider accelerometers and gyroscopes.

Accelerometers

Accelerometers measure the specific force acting on the sensor through inertial means. The simplest version of this is a test mass attached to a spring with a damper. When the sensor moves, the spring compresses or stretches due to the inertia of the test mass. By measuring the change in length of the spring, we can identify the acceleration on the mass, its sensor, and the vehicle it’s attached to. Additionally, by placing the mass and spring vertically, we can measure the acceleration due to gravity (more on that another time).

By plotting the acceleration over time, then integrating, we can determine the system’s change in velocity. Doing this again with the velocity allows us to determine the system’s change in position. By placing three accelerometers in each axis of motion, we can determine the system’s acceleration, velocity, and position at any point in time by tracking and integrating the measured accelerations.

Fortunately, accelerometers can be made far smaller than our spring mass example. You’ll often find them in MEMS packages, conveniently sized for integrated circuits to be used on a circuit card assembly.

Gyroscopes

Gyroscopes measure orientation and (most often) angular velocity. The most common example is a rotating disc mounted by three gimbals. Once the disc is set spinning, by conservation of angular momentum, it will tend to maintain its orientation. Attempting to rotate the apparatus will result in the disc maintaining its orientation while the gimbals rotate around it. This allows the user to measure the change in orientation from the reference provided by the disc.

While the larger mechanical gyroscopes are still used for highly accurate systems, nowadays MEMS gyros are ubiquitous. Beyond that, ring laser and vibrating structure gyroscopes also fulfill their own cost and accuracy applications. Most gyroscopes measure angular velocity rather than orientation directly, however, and we will proceed assuming our measurements are for angular velocity.

Much like with accelerometers, we can plot the angular velocity over time and integrate it to find angular displacement. Also similar to accelerometers, gyroscopes are described by number of axes measured.

With these two packaged into an IMU, we can determine our orientation and position using dead reckoning. However, obtaining accurate results can be far more difficult than it sounds.

Errors (Deterministic)

It can often be misleading to see the relatively small error rates and assume these sensors maintain high accuracy. But it’s important to remember that these sensors don’t measure displacement (angular or otherwise). Attitude and position are determined from the (successive) integrations of each measurement.

What if our accelerometer isn’t perfect, what if it has an error? Assume our accelerometer has an error of $± 1 mg$, where $g = 9.81 \frac{m}{s^2}$ so $± 0.00981 \frac{m}{s^2}$. Sitting on a desk, at $t = 10s$ our position is now $± 0.49 m$, our velocity is now $± 0.10 m/s$. At $t = 60s$, our position is $± 17.7 m$, our velocity is $± 0.59 \frac{m}{s}$. At $t = 1 hr$, our positional error now becomes $63.6 km$.

Error Accumulation Conceptualized

We can do the same with pointing error, however calculating positional error becomes more tedious as we would need to consider the pointing error over the distance travelled (e.g. a $5 \frac{m}{s}$ constant velocity with a $2 \frac{deg}{hr}$ drift over an hour would give a lateral error of a little over $5 m$, over $100 m$ with a $10 \frac{deg}{hr}$ drift).

Error Accumulation for Attitude Conceptualized

It should now be apparent that errors with accelerometers and gyroscopes on small time scales can be negligible but can accumulate to produce large inaccuracies. While actual error calculations are much more involved and nuanced, the principle remains the same.

Scale Factor Error

Scale Factor Error Conceptualized

The Scale Factor error identifies the error in the output measurement as a result of how the sensor scales the input event. This is normally given as a linear error percentage or ppm, though may also be accompanied by a nonlinearity scale factor. For instance, a gyroscope measuring a $5 \frac{deg}{s}$ rotation with a $0.5\%$ scale factor will output $4.97 \frac{deg}{s}$. Since Scale Factor error is often non-linear, linear approximations may create inaccuracies in larger measurements.

Orthogonality Errors

Due to manufacturing tolerances, multi-axis sensors can’t be aligned completely orthogonal to each other. They will always be sensitive to axes they shouldn’t be measuring.

The cross-axis sensitivity is the error caused by a sensor axis being sensitive to an input from a different axis.

Misalignment is the error resulting from misalignment of the sensor axes with the prescribed axes defined by the sensor package.

Misalignment Error Conceptualized

Acceleration Sensitivity

Ideally, gyroscopes would only measure angular rate. However, due to manufacturing tolerances and design, they are also sensitive to linear accelerations. This comes in two types: g-Sensitivity for bias shift subjected to linear acceleration (in both parallel and perpendicular axes) and g²-Sensitivity or Vibration Rectification, the bias shift due to oscillatory linear accelerations (in both parallel and perpendicular axes).

Uncertainty and Noise (Non-Deterministic)

By far the most important parameters that govern accelerometer and gyroscope use and selection are their parameters governing drift due to noise. Like all sensors, accelerometers and gyroscopes are not immune to noise. Given their safety critical applications and propensity to accumulate error, a lot of work has been devoted to identifying and determining noise affecting these sensors. We’ll focus on three characterizations of uncertainty due to noise: high frequency, lower frequency, and low frequency. It’s important to point out that we’re making a noticeable differentiation between error and uncertainty here. By uncertainty, we mean that it cannot simply be applied to the determination as a simple linear quantity. These parameters are formulated from signal theory and describe statistical quantities. For example, multiplying Angular Random Walk by the square root of time does not result in the error, but a single standard deviation capture of the particular noise.

Heading Drift/Uncertainty Conceptualized

The graphic above conceptualizes the uncertainty of the heading estimation from measurements of a gyroscope rigidly fixed to a table. As time progresses, the integration of noise on the output of the gyro adds uncertainty to our estimation, showing up as drift. Note that, as stated above, noise is non-deterministic and we cannot assume that the actual estimation deviates linearly per bias drift because it is a statistical property. The graphic is a simplified representation to illustrate the concept of drift.

Bias

You’ll often hear the term “Bias” used when referring to accelerometers and gyroscopes. Bias refers to the constant error between the true value and the measured value. This is important to know for calibration and compensation. For instance, if we take readings from a gyroscope that’s known stationary and they average 1.0 deg/second, we can offset future readings by this bias to more accurately measure our orientation. The results from these specific readings would be referred to as the Static Bias. Unfortunately, it’s not the only thing we have to account for.

Random Walk

Random Walk is the characterization of the uncertainty in the sensor output due to high frequency noise, attributable to white noise. Because of its high frequency nature, we expect the uncertainty from Random Walk to immediately impact our determinations.

Angle Random Walk (ARW) refers to gyroscopes and is given in $\frac{deg}{\sqrt{s}}$. Velocity Random Walk (VRW) refers to accelerometers and is given in $\frac{m}{s\sqrt{s}}$. The units of $\sqrt{s}$ can be confusing, but for now, just know that Random Walk is proportional to the square root of time.

Can we filter high frequency noise? Yes, that’s possible by playing with sampling. But it’s important to remember that filtering out high frequency noise could also miss important, quick event data (e.g. a car momentarily swerving).

Rate Noise Density

The white noise impact can be further clarified by the Rate Noise Density. This indicates the amount of noise present in the output within a sample frequency band. Think of it like the noise present per unit bandwidth frequency. Sampling at low frequencies (averaging more data points) will filter out more noise than sampling at higher frequencies.

How is it used? Random Walk is the integration of the Rate Noise Density over time at a sampling frequency of 1 Hz. So while Random Walk is an important parameter for general sensor specification, Rate Noise Density can be used for a more application-specific measure of uncertainty in place of Random Walk.

Let’s illustrate this with an example. A gyroscope with a noise density of $2.0 \frac{mdeg}{s\sqrt{Hz}}$, sampled at 100 Hz will have a noise standard deviation of $2 * \sqrt{\frac{100}{2}} = 14 \frac{mdeg}{s}$. If we integrate this over time, we get $0.85 \frac{deg}{\sqrt{h}}$. Notice that the units match the Angular Random Walk. But since ARW is determined from a $1 Hz$ sample, this value is not equal to ARW.

Bias Instability or In-Run Bias Stability

The Bias Instability characterizes lower frequency uncertainty due to noise than Random Walk, specifically pink noise. It’s given in $\frac{deg}{s}$ or $\frac{m}{s^2}$.

It’s often considered the most important quantity for identifying the accuracy of a sensor because it identifies uncertainty behavior over longer periods of time. Hence the idea of the instability of the bias (the offset between the error and true value). Though it’s important to remember that Random Walk uncertainty is also constantly accumulating (like all other uncertainties and errors) and should also be used when selecting a sensor.

Rate Random Walk

Rate Random Walk characterizes lower frequency uncertainty due to noise than Bias Instability, specifically brown noise. It’s given in $\frac{deg}{s^2\sqrt{Hz}}$ or $\frac{m}{s^3\sqrt{Hz}}$.

Rate Random Walk is probably used least often when selecting a sensor because of its low frequency (slow) affect on uncertainty, but is still nonetheless important for uncertainty estimation.

Interaction with Temperature

The noise characterizations are often determined from measurements of the sensor in ideal conditions: stationary, constant temperature. However, the noise (and consequently the characteristic parameters) will change with fluctuating temperatures. It’s important to keep this in mind since these parameters can change attitude and position uncertainty significantly.

Conclusion

There’s much more that can be said about these two sensors (and much more involved math), but let’s leave it there for now. Hopefully this overview has provided some perspective on how accelerometers and gyroscopes can be used to estimate position and attitude, and the important parameters when selecting them for applications.