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Kalman filtering of IMU data

Introduction

To many of us, kalman filtering is something like the holy grail. Indeed, it miraculously solves some problems which are otherwise hard to get a hold on. But beware, kalman filtering is not a silver bullet and won’t solve all of your problems!

This article will explain how Kalman filtering works. We’ll use a more practical approach to avoid the boring theory, which is hard to understand anyway. Since most of you will only use it for MAV/UAV applications, I’ll try to make it look more concrete instead of puzzling generalized approach.
Make sure you know from the previous articles how the data from “accelerometers” and “gyroscopes” are used. Some basic knowledge of algebra may also come in handy :-)

Basic operation

Kalman filtering is an iterative filter that requires two things.
First of all, you will need some kind of input (from one or more sources) that you can turn into a prediction of the desired output using only linear calculations. In other words, we will need a lineair model of our problem.
Secondly, you will need another input. This can be the real world value of the predicted one, or something that is a good approximation of it.
Every iteration, the kalman filter will change the variables in our lineair model a bit, so the output of our linear model will be closer to the second input.

Our simple model

Obviously, our two inputs will consist of the gyroscope and accelerometer data. The model using the gyroscope data looks like this:

The first formula represents the general form of a linear model. We need to “fill in” the A en B matrix, and choose a state x. The variable u represents the input. In our case this will be the gyroscope’s data. Remember how we integrate? We just add the NORMALIZED measurements up:

alpha k = alpha k-1 + (_u_ kbias)

We need to include the time between two measurements (_dt_) because we are dealing with the rate (degrees/s):

alpha k = alpha k-1 + (_u_ kbias) * dt

We rewrite it:

alpha k = alpha k-1bias * dt + u k * dt

Which is what we have in our matrix multiplication. Remark that our bias remains constant! In the tutorial on gyroscopes, we saw that the bias drifts. Well, here comes the kalman-magic: the filter will adjust the bias in each iteration by comparing the result with the accelerometer’s output (our second input)! Great!

Wrapping it all up

Now all we need are the bits and bolts that actually do the magic! These are some formulas using matrix algebra and statistics. No need right now to know the details of it. Here they are:

 u = measurement1 Read the value of the last measurement x = A · x + B · u Update the state x of our model y = measurement2 Read the value of the second measurement/real value. Here this will be the angle calculated from our accelerometer. Inn = y – C · x Calculate the difference between the second value and the value predicted by our model. This is called the innovation s = C · P · C’ + Sz Calculate the covariance K = A · P · C’ · inv(_s_) Calculate the kalman gain x = x + K · Inn Correct the prediction of the state P = A · P · A’ – K · C · P · A’ + Sw Calculate the covariance of the prediction error

The C matrix is the one that extracts the ouput from the state matrix. In our case, this is (1 0)’ :

alpha = C · x

Sz is the measurement process noise covariance: Sz = E(zk zkT)

In our example, this is how much jitter we expect on our accelerometer’s data.

Sw is the process noise covariance matrix (a 2×2 matrix here): Sw = E(x · xT)
Thus: Sw = E( [alpha bias]’ · [alpha bias] )

Since only the diagonal elements of the Sw matrix are being used, we’ll only need to know E(alpha2) and E(bias2), which is the 2nd moment. To calculate those values, we’ll need to look at our model: The noise in alpha comes from the gyroscope and is multiplied by dt2. Thus: E(alpha2) = E(u2)· dt2.

These factors depend on the sensors you’re using. You’ll need to figure them out by doing some experiments. In the source code of the autopilot/rotomotion kalman filtering, they use the following constants:

E(alpha2) = 0.001
E(bias2) = 0.003
Sz = 0.3 (radians = 17.2 degrees)

Gyroscope to roll, pitch and yaw

Now you know most practical things you need to know about accelerometers, we’ll continue with gyroscopes (other names: gyro, angular rate sensor).

A gyroscope is a very fancy name for a device that measures the angular rate (how much degrees per second it is rotating). The gyroscopes used in very critical applications (like a jumbo jet) are very advanced and complicated. Fortunately for us, there are some low-cost and small sized alternatives which are good enough. They are fabricated using MEMS (iMEMS) technology by big companies like Analog Devices.

As you probably remember from you physics class, position, velocity and acceleration are related to eachother: deriving the position, gives us velocity:

d x = v x

with x being the position on the x-axis and v x being the velocity along the x-axis.
Maybe less obvious, the same holds for angles. While velocity is the speed at which the position is changing, angular rate is nothing more than the speed the angle is changing. That’s right:

d alpha = angular rate = gyroscope output

with alpha being the angle. It’s starting to look pretty good! Knowing that the inverse of deriving (d .) is integrating (∫), we change our formula’s into:

angular rate = ∫ gyroscope output = alpha

Woohoo, we found a relation between angle (attitude!) and our gyroscope’s output: integrating the gyroscope, gives us our attitude-angle.

Enough boring theory! Let’s take a look at some figures. The following figures all represent the same motion: I took a gyroscope, turned it 90 degrees left and back, and turned it 90 degrees right and back.

The raw data (used here) is what we get when we feed the gyroscope’s output (0-5 volt) into a 10-bit ADC (analog to digital convertor). So the raw values are between 0 and 1024. Here’s a figure of that:

(The red line is just a low-pass filtered version of the blue data)
You can clearly see a positive angular rate followed by a negative one. But we’ll need to shift the figure down, to make sure negative values correspond to a negative angular rate. Otherwise the integration (which can be seen as the sum of our y-values) would keep adding up values and never substracting any! We normalize it by substracting about 490 from every value. This normalization gives us the following figure:

Now all we need to do, according to our formulas, is integrate it! Some of you may still have nightmares about college and start shivering when they hear the word integration, but its pretty simple. Discrete integration is nothing more than summing up all the values! Basically, integration from 0 to the i^th^ value:

integration(i) = integration (i-1) + vali

This is the simplest possible integrator. A more advanced one, which also flattens out possible jitter in the data, is the runge-kutta integrator:

integration(i) = integration(i-1) + 16 ( vali-3 + 2 vali-2 + 2 vali-1 + vali)

Using this runge-kutta integration, we get the following figure:

This is pretty much the exact movement I made! Now we just need to add a scale factor to our data so our result is in degrees:

This pretty much ends my story of the simplified gyroscope!

In reality, gyroscopes are suffering from an effect called drift. This means that over time, the value a gyroscope has when in steady position (called bias), drifts away from it’s initial steady value:

The blue line gives you an idea about the drift. During 4500 samples (12 seconds in my setup), the bias drifted about 30 degrees!

Remember that we need the bias (about 490 in our example) to normalize our data. How can we integrate when we have no idea about the currect bias? We’ll need to find a way to get it. More about this in the next article. A hint: our accelerometer isn’t affected by drift ;-)

Accelerometer to pitch and roll

This tutorial descibes how an accelerometer can be used to determine an aircraft’s attitude (pitch and roll).

An accelerometer measures, as it’s name hints, acceleration along a predefined axis. As you probably remember from you physics class, the earth’s gravity is also an acceleration (a falling stone keeps going faster and faster). So: with an accelerometer, we can measure the earth’s gravity! This image shows how we do it:

The red arrow represents the earth’s gravity. The blue arrow shows how the accelerometer senses gravity. Note that the axis of this accelerometer is perpendicular to the aircraft (we placed it like that in our aircraft!).
The angle theta between the actual gravity vector and the measured gravity is related to the pitch of the aircraft (pitch = theta + 90°). If we know theta, we know our pitch! Since we know the magnetude of the earth’s gravity, simple calculus gives us our pitch angle:

accelerometer = cos (theta) * gravity

theta = acos (accelerometer / gravity)

And since pitch = theta + 90°

pitch = asin (accelerometer / gravity)

Woooow, we calculated the pitch orientation of our airplane using an accelerometer. Pretty easy, huh?

Calculating the roll angle is pretty much the same. We only need an extra accelerometer with an axis perpendicular to the pitch-accelerometer.

Reality is a bit different from this simplified example. The inverse sinus can’t give you the full 360 degrees ranging pitch angle. A plane heading for the sky and one heading for the ground would both result in a 0 (zero) measurement. We’ll need an extra accelerometer to distinguish these cases. The 2-argument inverse tangens makes sure the resulting angle is in the correct quandrant. Thus:

pitch = atan2(accelerometer / gravity, z / gravity)

Now you know most about using accelerometers to calculate pitch and roll, don’t start building your own autopilot system just yet! There are more forces working on a flying airplane then just good old gravity! Just think about the centripetal force when following a circle path. We’ll need gyroscopes to correct this over short period of time (also usefull to eliminate the effect of vibrations on the accelerometer). Over a longer period of time, we’ll need some more advanced physics to estimate these other forces so we can compensate for them. I’ll write later about the different approaches you can use to do this.
The gyroscopes are covered in the next tutorial: gyroscope to roll, pitch and yaw

What is an IMU

Every good autopilot system needs an IMU-module. IMU stands for Inertial Measurement Unit. Basically, you can look at it as a black box that gives information concerning your current position and orientation. Position could be GPS-coordinates. Orientation could be roll, pitch and yaw. Sometimes you need to calculate it yourself, sometimes the black box calculates it for you. Soooo, since an autopilot needs to know the current position and orientation, it needs an IMU.

 The 6DOF by SparkFun electronics. Notice the 2 gyroscopes standing in upright position. One for each axis!

Every interested mind now wonders “how does this black box work?”
Well, I’ll give you the three most importent sensors.
The first one is, of course the GPS. It gives you your current position and your speed.
The second one is a set of 3 accelerometers. Each one senses acceleration in one direction. Remember from your physics class that gravity is also an acceleration! So the 3 accelorometers give you the 3 components of the gravity vector! Knowing that the gravity vector is supposed to point right to the middle of the earth, you can calculate the orientation! Keep in mind that when an MAV is flying, other forces like centripetal acceleration may badly influence those 3 accelerometers used to sense the gravity vector. As a result those measurements are only correct when averaged over a longer period of time.
The last one is a set of 2 or 3 gyroscopes. Thats a very fancy word for a sensor that senses the speed of rotation (also called angular velocity). With 3 gyroscopes you can sense the speed of rotation around your 3 (x, y, z) axes. Clever readers may notice that the mathematical integration of the values give you the orientation (integration of speed = position)! Unfortunately, there is a lot of drift on those values, so they are only correct for a short period of time…

Readers who payed attention noticed that the orientation given by the accelerometers are correct over a long period of time, and the one given by the gyroscopes are correct over a short period of time. Can’t be combine those two?! Yes we can! A very special filter called the kalman filter does the trick.
Nowadays, every guided vehicle (from a guided missile to a jumbojet) uses those sensors with the kalman filter. Mr. Kalman who invented his filter in the fifties must be so proud!

5 May 2006, 14:24 | Link | Reacties

Overview: autopilot modules

I tried to list the most interesting autopilot systems.

Name Company Remarks
Paparazzi / A DIY project
AFCSV2.5 Rotomotion For heli’s
MP2028^g^ MicroPilot 28gr incl. GPS!
Phoenix o-navi Not including software?
TGE C. C. & design
Piccolo Cloud Cap Tech
PicoPilot uNav Mostly hobbyist-oriented?

MP2028^g^

MicroPilot is a company solely focused on miniature autopilot systems. They also offer various add-on modules.

Phoenix

The o-navi company is specialized in navigation modules using MEMS sensors. Their top product is an integration of their IMU module in a full flight controller engine: the Phoenix

Tiny Guidance Engine (TGE)

Continental Controls & Design is a small company with one product: a small autopilot system. This includes the IMU-module with microcontrollor and ground control software. Their main client is the US army.

Piccolo

Cloud Cap Technology is a company with UAV-autopilot related modules as only interest. Their autopilot look very professional but big and heavy, compared to their IMU-modules.

Parapazzi

This project aims to develop an autopilot under the GNU public license. While not that advanced (eg. stabilization is done using 4 thermophile sensors), it has proven to be very succesfull.

Rotomotion

The roots of the rotomotion company lie in the autopilot project on sourceforge. Now, they sell a more advanced version of this project. As a plus, they also offer fully equiped UAV helicopters.

PicoPilot

This company offers a very lightweight autopilot solution as a composition of various modules. It looks oriënted at the advanced RC-hobbyist.

5 May 2006, 13:46 | Link | Reacties [4]

Overview: commercial IMU's

I tried to list the most interesting commercial available IMU-module. These can be interesting if you plan to develop your own autopilot(-software).

Name Company Size Weight Remarks
TrIMU Mavionics 40×40×14 15g
Atair INU Atair 38×50×19 45g
GyroCube o-navi 38×32×15 8,2g
uNAV Crossbow 57×45×26 33g Open source software
3DM-GX1 MicroStain 42×40×15 26g
TGE C. C. & design 30×30×18 13g Full autopilot?
Crista Sensor Cloud Cap Tech 28×29×15 8,5g
MP2028 MicroPilot 100×40×15 28g Full autopilot
6 DOF Sparkfun El. 51×51×23 21g With bluetooth link
MTi Xsens 58×58×22 50g RS232 & USB

I know this list is far from complete. Please contact me if you know other commercial available IMU’s.

5 May 2006, 12:41 | Link | Reacties [2]

Gas

Johannes van Helmont in “Ortus medicinae” (Oorsprong van de geneeskunde), 17e eeuw

Hunc spiritum, incognitum hactenus, novo nomine Gas voco, non longe a Chao veterum secretum.

Voor de niet klassiek geschoolden onder ons: “Ik geef deze tot nog toe onbekende geest de nieuwe naam Gas, naar analogie met Chaos van de Oudheid”.
Ofte hoe één persoon op één dag een woord kan uitvinden dat hedentendage in vele talen gebruikt wordt.

6 May 2005, 11:22 | Link | Reacties