## Unmanned Systems: MEMS Inertial Sensors and Vibration

We wanted to have a discussion on MEMS inertial avionics and how improper sensor selection, and integration methods can destroy the validity of the output from such precision devices.  Most small UAVs today have only one IMU (inertial measurement unit typically consisting on a triad of accelerometers and a triad of rate gyros) and if this IMU is not outputting excellent measurements in all axes, your vehicle will not be providing for adequate navigation required for automatic control.  After all, we need to have precise navigation/state estimation in order to control precisely.  To compound the problem, it can be difficult to determine the sensors and circumstances leading to the production of erroneous data.  All too often the controls engineer or the control design itself, be it optimal, robust, adaptive, classical, or some combination thereof is blamed for poor control when in reality, the navigation itself contains the significant errors which lead to poor controller performance.

One of the problems frequently encountered is in the area of vibration rectification.  Vibration rectification is an effect where vibration can cause a bias shift in accelerometer output, typically because of asymmetries in the positive and negative accelerations. What’s that you say?  Accelerometers designed for inertial measurement of aircraft often require excellent bias stability.  This is because we are highly interested in the direction and magnitude of the gravity vector in order to approximate the aircraft Euler angles.  This may assume you are interested in attitude feedback control.  OK, so what’s the problem, I’ll just select an accelerometer with excellent bias stability, right?  Not exactly.  Due to the low power and small form-factor requirements of today’s small UAVs, MEMs accelerometers are often used in small UAV IMU applications.  What’s the problem?  Most of the available MEMs accelerometers are deficient for the purpose of inertial navigation.  This is true even of many accelerometers being used for inertial sensing today.  This is true even for MEMS inertial sensors designed and marketed for the purpose of inertial sensing!  Several capacitive type MEMS accelerometers  have proven themselves worthy of the challenge, but even specialty capacitive MEMS accelerometers designed for inertial measurement have nonlinearity problems which can cause significant error under certain dynamic conditions, specifically certain vibration.

The nonlinearity present can be significant when there is a static term combined with an oscillating term on the acceleration being measured capacitively, ie. vibration.  The good news is that you don’t have to worry about all of this if your vehicle has no vibration.  The bad news is that vibration is everywhere and even small amplitude vibrations can be very problematic at certain frequencies.  In the end, under certain vibration environments, a dc offset signal (which will cause significant errors in the rectified attitude estimations) can be generated from the vibration signal all due to the nonlinearity of the measurement of the MEMS sensor:

$\frac{\Delta^2}{d^2} \approx \frac{\Delta^2 sin^2(\omega t)}{d^2} \approx \frac{\Delta^2(1-cos(\omega t))}{2d^2}$

These nonlinearities may be mitigated by subtract them using various differential capacitor arrangements.  The squared nonlinear terms can then be eliminated and only the cubic terms and higher are consequential and these terms are at least significantly smaller than the squared terms.

Big deal, I’ll just select a sensor that specifies a low vibration rectification coefficient, right?  That’s right! Although as you can imagine, it is difficult to get performance and bias stability data on many of today’s inertial sensors with respect to their performance throughout the vibration environment you are interested.  This is especially true when dealing with today’s advance VTOL unmanned aircraft concepts where every frequency imaginable is present, aliasing other frequencies and making your PSD/FFT plots look like a porcupine.

To prevent erroneous or spurious data from presenting themselves on the inertial avionics, various “rules of thumb” and other experimental methods are common but not necessarily practical.  You may have seen various techniques being used in labs today to mitigate high frequency vibration on IMUs such as wrapping the IMU in foam.  (This technique is a bit silly as you can imagine that it will be nearly impossible to keep the IMU’s reference with respect to the airframe constant.  Furthermore, foam may collect moisture, fuel, dirt, as well as it may wear resulting in changing its “filtering” properties.) Other, more practical techniques include selecting appropriate vibration isolators, of which there are many varieties.

Repeatable integrations of inertial avionics are obviously preferable and design work should be performed to specify a suitable vibration isolation system that at a minimum is: consistently repeatable on multiple airframes, is robust and cannot easily fail, and one that exhibits high attenuation at frequencies beyond those that you are interested in sensing.   It should be clear that there is no such thing as a generic integration design suitable for all inertial avionics and on all UAVs.  It is up to the designer to understand all of the magnitude & frequency content that the inertial sensor will be subject to, and which of this content must be destroyed in order to not offend the sensor you have chosen.

There are numerous validation methods that can be used to verify your vehicles navigation.  The point is this validation needs to be performed prior to testing controllers in flight.  Otherwise, you may discover control problems (poor closed loop damping) and then not know if the problems stem from navigation or control.

-Lance Holly

## Gyro Drift: Why You Can’t Walk in a Straight Line with Your Eyes Closed

The other day I was watching Mythbusters and saw something that comes up often when discussing flight control systems. The show, which aired on October 12, was named “Walk a Straight Line.” The premise of the myth is that people cannot walk in a straight line when they don’t have a point of reference, such as when they are blindfolded. I think everyone has some experience with this as we have all tried walking through a dark room inevitably bumping into any number of shin-height objects.

The show went on to prove that, indeed, it is difficult, if not impossible to navigate with your eyes closed. The first thing that piqued my interest in the myth is that the reason for our navigational difficulty when blindfolded is similar to what happens to an unmanned aircraft when you try to navigate with rate gyros alone. Essentially, when we close our eyes, we now rely on our internal sensors, which can detect movement, but not position. For example, if you sit in a chair, close your eyes, then let someone spin you slowly. You will be able to tell which direction you are spinning, and roughly how fast you are spinning. If you combine these two pieces of information with the amount of time you were spinning you can arrive at an estimate of how far you have been rotated. For the mathematically inclined you just integrated your rotational rate through time to estimate the angle you rotated as is shown here:

$\theta = \int_{T_1}^{T_2} \omega dt = \omega(T_2-T_1)$ for constant $\omega$

This is essentially what we are doing when trying to blindly walk in a straight line: constantly use our “rate sensor” to estimate how far we have turned off course. The problem is that our internal rate sensor, much like the rate gyros in a UAV, have some bias error. When we introduce this bias error into the above equation we get:

$\theta = \int_{T_1}^{T_2} \omega + error dt = \omega(T_2-T_1) + error(T_2-T_1)$ for constant $\omega$ and constant error

So when we introduce an error in the rate measurement and integrate through time, we see that the error in our attitude estimate grows with time. This is why Jamie and Adam diverged more and more the further they walked. You might be asking, “Is the error really constant?” The answer is no, but it is typically biased in one direction. This explains why Adam constantly veered to the left.

So how do we deal with this rate bias? You can improve the sensor to minimize the bias as is done in fiber-optic gyros and ring-laser gyros, or you can add another sensor that corrects for the bias. In people this second sensor is our eyes. In aircraft this sensor can be a magnetometer for heading corrections, or accelerometers for pitch and roll attitude correction. Getting these sensors to work together can be a trick, which is where the wonderful Kalman filter comes into play. I won’t get into any details of Kalman filtering, except to say that it is a method of combining multiple measurements to arrive at the best guess of a systems current state.

There was one more part to the myth that was really interesting. At the end of the show, Jamie and Adam linked themselves together with a ladder while they walked. The concept was that they would cancel the error of the other person and walk in a straighter line. The results weren’t great, but the concept does hold promise. If you build an inertial measurement unit for a UAV using multiple rate gyros in each axis and average their outputs, you can actually get better results.

The last thing I wanted to mention is that at one point in the show Jamie and Adam put buckets on their head to simulate poor visibility. I have been through survival training in Antarctica and this is definitely part of the course. They call it the bucket-head excercise. It is meant to simulate white-out conditions. It is shocking how disoriented you can get when you can’t see your surroundings. There are multiple methods people use to navigate in these scenarios, but the most interesting is what the New Zealand teams do. They use the wind as a type of compass. If you turn until the wind is at your back (or your side or face, etc), then you can use this information to get some amount of orientation information assuming the wind isn’t changing directions. Pretty interesting.

-Bill Donovan