# Arduino IMU: Pitch & Roll from an Accelerometer

One day, looking for cheap sensors on ebay, I found this interesting board which contained everything I was looking for. It basically consists of a 3-axis accelerometer (ADXL345), a 3-axis magnetometer (HMC5883L), a 3-axis gyroscope (L3G4200D) and a barometric pressure sensor (BMP085). My plan is to build an Inertial Measurement Unit (IMU) (or maybe I should call it Attitude and heading reference system (AHRS)) and in the process learn how to interact and interpret the information all of this sensors provide. The fact is I have some experience using IMUs since I used one on my master thesis and another one on the Avora AUV, but the fact is they come preprogrammed and there is not much point in working with the raw sensor data unless you want to improve the measurement or give it another use.

For this project I am also using an Arduino Duemilanove, for that reason I wanted to call it ArduIMU, but there is already another project with the same name, so I will have to find another name (suggestions would be appreciated). Connecting the sensor board to the Arduino is pretty straightforward, every sensor has an I²C interface so you can access each of them using the Arduino Wire Library. The drawing was done using fritzing, on which I created the corresponding custom part for this board, although I did something wrong and it does not conform to the fritzing graphic standards.

This will be the first of a series of posts I plan to write about this project, since there are several steps I need to take in order to fully understand each sensor and several more to combine them in order to improve accuracy. In this post I want to talk about the accelerometer and how to obtain the roll and pitch angles from it, which is a process that can also be called tilt sensing.

Accelerometers are devices that are capable of measuring the acceleration they experience relative to free-fall,  the same acceleration living beings feel. As a consequence, accelerometers are incapable of measuring the acceleration of gravity, but can be used to measure the upwards acceleration that counters gravity when at rest. This acceleration is measured as $1g$ on the z-axis, when both pitch and roll angles are zero, but when the sensor is tilted either the x-axis or the y-axis experiences a component of the upward acceleration, whose magnitude depends on the tilt angle.

### Pitch & Roll estimation

Obtaining the pitch and roll angles is then a matter of being able to read the accelerometer, convert these readings to the g unit (1g = 9.8 m/s²), and apply the corresponding equations. The process of obtaining and converting the accelerometer readings depends on the accelerometer you are using, in my case, the ADXL345 in its basic configuration, provides 10-bit resolution for ±2g, but has several other ranges (±2g, ±4g, ±8g, ±16g)  and resolutions (from 10 to 13 bits depending on the range) . Generalizing, the formula used to calculate the acceleration from the accelerometer readings is:

$G_{Accel} = Raw_{Accel} \cdot \dfrac{Range}{2^{Resolution - 1}}$

Once we have the correct acceleration components, we can proceed to calculate the different angles using the following equations:

$pitch = \arctan{\left(\dfrac{G_y}{\sqrt{G_{x}^2 + G_{z}^2}}\right)}$     $roll =\arctan{\left( \dfrac{-G_x}{ G_{z}}\right)}$

For more information about where these equations come from, you can read the documentation I include at the end of this post. As you can see, the denominator of the pitch equation is defined to be always positive, so the equation itself only provides $[-90, 90]$ range, which is exactly what is expected for the pitch angle. In contrast, the roll equation provides $[-180, 180]$ range. It is important to take into account that when the pitch angle is 90º, the surge axis (roll) is directly aligned with the gravity vector, thus we cannot measure the roll angle anymore, this is what is called Gimbal Lock.

Also, be aware that the roll equation is undefined when both $G_x$ and $G_z$ are equal to zero, and that for each possible value of the calculation done inside the arctan function there are two valid solutions, not only on the roll but also on the pitch equation. These problems can be easily solved in code by using the function atan2, which eliminates the angle calculation ambiguity by taking into account the quadrant.

### Removing short-term fluctuations using a Low-Pass filter

At this point we already have a fully functional pitch & roll estimation system, but if we experiment with it we will discover that the readings fluctuate quite a bit and this may be very annoying for some applications. Removing these short-term fluctuations can be achieved by means of what is called a Low-Pass filter. This type of filter attenuates the higher frequencies of the signal, thus providing a smoother reading. The Low-Pass filter is easily implemented by using the following equation:

$y_{t} = \alpha \cdot x_{t} + (1 - \alpha) \cdot y_{t - 1}$

Where $y_t$ is our filtered signal, $y_{t-1}$ the previous filtered signal, $x_t$ the accelerometer reading and $\alpha$ the smoothing factor. It probably may seem obvious, but filtering should be done to the accelerometer readings before calculating the angles, instead of to the angles themselves. Regarding the smoothing factor, the lower we set it, the more it will take for the angle to stabilize, so we should not set it too low because then we could lose real-time behaviour. With this I mean that the reading will not correspond to the real angle until it stabilizes, and this could take some time.

### The source code & the ADXL345 library

I developed a small library to interface with the accelerometer, even though at the moment I have only implemented the basic functionality, I plan on supporting all of the device features. You can find it in my gitlab account. Thanks to the library, the code is pretty straightforward. It just reads the sensor accelerations which are already converted into gs by the library, applies the Low-Pass filter and then uses the roll and pitch equations to calculate the angles.

#include
#include

const float alpha = 0.5;

double fXg = 0;
double fYg = 0;
double fZg = 0;

ADXL345 acc;

void setup()
{
acc.begin();
Serial.begin(9600);
delay(100);
}

void loop()
{
double pitch, roll, Xg, Yg, Zg;
acc.read(&Xg, &Yg, &Zg);

//Low Pass Filter
fXg = Xg * alpha + (fXg * (1.0 - alpha));
fYg = Yg * alpha + (fYg * (1.0 - alpha));
fZg = Zg * alpha + (fZg * (1.0 - alpha));

//Roll & Pitch Equations
roll  = (atan2(-fYg, fZg)*180.0)/M_PI;
pitch = (atan2(fXg, sqrt(fYg*fYg + fZg*fZg))*180.0)/M_PI;

Serial.print(pitch);
Serial.print(":");
Serial.println(roll);

delay(10);
}


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## 75 Comments

1. Very clear presentation! Just a comment that there is a noticeable delay between you moving the accelerometer and the cube image moving. This could significantly detract from the agility of a robot or game player using this system. Considering that the mechanics processing required is negligible, this leaves the sensor latency, smoothing filter and graphics processor as possible sources of delay. Could this lag be reduced by assuming continuity using something like Taylor series prediction?

1. Anil says:

I would have to look into it, but it probably can be done if you assume continuity. For most applications I can think of, it wouldn’t be very useful though.

2. Rey says:

How about calling it ” ArduIMU “

1. Anil says:

Thank you for your suggestion, although that name is already used in a similar project.

3. David Gonzalez says:

hey buddy.. here’s my contribution, how about calling it imuino, you could use the emu bird as a logo or am I going too far with it?

1. Anil says:

I like the name, thanks!

But I’m not sure about the bird…

1. David Gonzalez says:

hahaha.. I understand!!!

4. Hi , i have the exact same sensor (you can buy it for 20bucks on dx.com) , but i didnt start to work on it yet , but your tutorial will be really useful, thank you .
i have a question thou ,
can you get SWAY SURGE and HEAVE “like” data , by subtracting the angle from the gyro to data of the accelerometer , and use the result to get translation (or more like the position relative to the starting point), with doing something like x+=x; y+=y; z+=z; (in the loop) with the result ?
? (not sure if i am clear , i am a beginner).
you could use something with DOF in your lib name (Degree Of Freedom) .
Cheers

1. Anil says:

Thanks! Well, the gyro doesn’t really give you an angle but an angular rate. Giving that you don’t have translational speed, nor is it calculated anywhere, you can’t really measure translation. What you are asking falls more within the grounds of an Inertial Navigation System (INS), and in itself is very a complicated problem, but an inaccurate approximation could be done by integrating over time the acceleration.

1. i am getting confused , (it is quite out of my league but it is so fascinating!)
the accelerometer gives the acceleration (relative to gravity) (if not translating , gives roll and pitch)
the gyro gives the angular rate (=angular velocity=speed of rotation ; different from the acceleration of the rotation ?) (if integrated gives the orientation ?)
but then the magnetometer (understand compass) gives the orientation ,right ?

so all the youtube videos of people doing the Euler thing (orientation tracking) with 6DOF sensor , are basically doing the work of the magnetometer ?

if i want to do the integration over time (extract translation and find position) (solving the shifting with a regular manual reset/recalibration) , i would need to remove the pitch yaw roll from the accelerometer data resulting only in the translation acceleration , right ?

also this is relevant http://www.youtube.com/watch?v=6ijArKE8vKU

do you want we continue this “talk” on a forum ? i don t want to “spam” your comment space.

1. Anil says:

First of all, the accelerometer gives yo the acceleration relative to free-fall, so can actually extract a “gravity” vector from it when at rest, that may have a component on the x, y and z axis. Since the gravity vector es pointing up because it’s not really gravity but the acceleration that counters it, you can only extract tilt from it, i.e. pitch and roll.

The gyroscope gives you the angular rate, which I believe it’s equivalent to angular velocity, certainly not to acceleration. So if you integrate over time you can obtain the orientation, but not quite, it’s something I’ll talk on the next post about the Arduino IMU.

Magnetometers measure magnetic field in each axis, so you can “sense” the earths magnetic field and calculate the orientation relative to the magnetic north (yaw), but there are techniques to calculate pitch and roll also. I will talk about it in future posts.

I don’t really know about the videos you’re talking about, but calculating precise orientation usually requires a combination of sensors.

Regarding the translation measurement, you can extract the gravity vector from the accelerometer readings, so all you have left is the acceleration you need for calculating the translation. But you must remember that integrating over time accumulates a noticeable error and, in this case, you must integrate twice (acceleration->speed, speed->translation).

5. fdsa says:

There is a gyro on that board too, mix it with the accelerometer. Combine the outputs with a complementary filter first. Fool with the crossover points while graphing the outputs, adjust until you get an acceptable error from the fastest response you want to perform in. After that, you can start playing with extended kalman filters, they do a great job of mixing the best way possible, but they might run a little slow.

Nice job.

6. AbilFida says:

Hi everybody, could you please tell me about formula of yaw estimation ?
I mean that, there is written that to get roll from adxl345 is arctan(-Gx/Gz), so what about yaw ?
I hope understand later, thanks.

1. Anil says:

There is no way to extract yaw from an accelerometer.

7. NadirHdz says:

Hi, i have a problem with my sensor, it seems that it only shows one value for pitch and for roll..

This is my output in the Arduino Serial
-34.09:135.67

I only get to print something when SDA and SCL is disconnected, when I connect the wires, the serial stops.

Any idea why? Is my gyro damaged?

I am using an Arduino Mega

1. NadirHdz says:

any idea why? anyone?

1. Anil says:

Well, if you could tell us more about what you are doing and using and how it’s all connected…

1. NadirHdz says:

I am using the library that you posted here. All I want is to make it work and display the pitch and roll.. it is connected to 5V-GND and the SDA and SCL to pins 20/21 in the Arduino MEGA

2. I’m having the exact same problem, except that I’m using an Arduino UNO instead. Did you make it work? Great article, BTW!

8. Marcelo Nogueira says:

Nice post! Really liked it! Helped me a lot. Congrats!
Just one thing: isnt the Low-Pass filter equation with a sign error? Shouldnt it be alpha*x + (1-alpha)*y ?

1. AnilM3 says:

Fixed

9. paul says:

Hi found this very interesting I’m going to convert this to python and give it a go in the next couple of days it actually looks pretty simple. Any chance of getting the next part up on the gyro?

10. Anil says:

Yes, I will! I’ve just had two pretty busy months and haven’t had much time. Thanks for following and please be patient.

1. paul says:

cool thanks!

1. paul says:

Thought i’d post a quick message I implemented your code in python on my Raspberry Pi and am seeing very good results. I do get some odd noise on occasion which will give me a 2g spike (in a +-4g range) on the Z and X axis but never Y! Not sure why. I’m currently writing out to a txt file and then importing the results into excel for graphing. Now i’m happy my code is working ok i’m going to start creating a real time GUI. Not bad for my first bit of programming.

1. Anil says:

I’m happy it worked, thanks for writing! I hope in less than a month I’ll be able to write the next step on this project. I’m actually two/three steps ahead, but I don’t find the time!

11. Great job and a great post. Did you ever post your source code for the cube visualization? I’m very interested in doing something similar, so any guidance would be appreciated.

1. Anil says:

Thanks, you can find a link to the source code at the end of the post.

12. This is the perfect site for anyone who would like to understand
this topic. You understand so much its almost tough to argue with you (not that I actually
will need to…HaHa). You definitely put a brand new spin on a
topic which has been written about for years. Wonderful stuff, just great!

13. You have observed very interesting points !
ps nice site. “Formal education will make you a living self-education will make you a fortune.” by Jim Rohn.

14. m00se says:

Hi, Nice project, I have the exact same sensor and want to start playing with it.
I have been to your github and downloaded the zip, I renamed the arduino folder as suggested, opened it in arduino , and it wont compile,
I get the following message :
“pitch_roll:20: error: ‘ADXL345’ does not name a type
pitch_roll.cpp: In function ‘void setup()’:
pitch_roll:24: error: ‘acc’ was not declared in this scope
pitch_roll.cpp: In function ‘void loop()’:
pitch_roll:33: error: ‘acc’ was not declared in this scope

do you know what it going wrong ? from my knolwege of arduino the ACC function is not being recognised, but I have the folder correctly named with all files inside it and in the library folder,

thanks for any help , I look forward to playing with this

1. Anil says:

You need to put the renamed arduino directory in your workspace under another directory called libraries. If this directory doesn’t exist you should create it. Then open the example and try again. Cheers.

1. m00se says:

hey ,
I got it working last night, turns out there is a bug with arduino and you need to open the file in a certain way or it wont compile,
did you ever integrate the gyro into it ? I have tried the multiwii code with this IMU and its a lot smoother , although not as easy to get the attitude values out of,

so far im happy, 🙂

1. m00se says:

sorry , i meant to say thanks too, good work ,
do we know what angle presicion this gets?

15. which software you use for the 3D real-time? can provide the code and the setting? thanks….

1. Anil says:

It’s all explained in the post, including the code.

16. Andrei says:

Hi, you are using +-2g range on the accelerometer right? So, you should get +-512 radings no? That is what i’m getting but the pitch and roll angles have really bad readings like 12453 or even 23875.
What is the problem?
Please help me!

1. Andrei says:

I was able to resolve the problem…I don’t know what it was, I just started from the beginning and it worked from the second try. Thanks very much for your project it was of great help.

1. Anil says:

You’re welcome, glad you were able to solve it!

17. Cagdas says:

Hi. I am working one week for How can I calculate yaw and use gyro ? I can find roll and pitch but not find yaw. I work with ROS and rviz only accept quaternion. IMU is MPU6050 6DOF. Some people say degree = angular celocity * sample rate. But I try a lot of way but I do not do. How can I use gyro and find yaw ? Thank you good work.

18. Skad says:

Hi,
I’m working with IMU.
I need to define, every istant, the rotation matrix to do a coordinates changes from sensor coordinates system to a fixed system defined by the initial sensor orientation.
I have also the gyroscope data. But knowing the pitch, roll and yaw angles there are 6 rotation matrix build changing the order of the three rotation. How can I solve this problem?

Thanks

19. Very rapidly this web page will be famous among all blogging people, due to it’s
pleasant content

20. mr.Kim says:

Good afternoon.
I am a student in Korea. For not proficient in English Thank you for your understanding.
Google translator was used.

Well I saw a great movie!

Time of graduation, work with Unity & Arduino is trying to create work.
Accelerometer use here but I can not think of what to do to try. not gyro..
It was found during a Google search you gonna be okay?
MMA8452Q Accelerometer sensor was used.

MMA8452Q XYZ values ​​received by vector3 (forward, left, rifht) will try to change.

Please respond. Thank you.

21. guest says:

Hi!

I’m so excited that i found this post! i’m a beginner interested in developing navigation/control of rov/auv. Please can you guide me how to proceed after i have an imu that gives magnetic heading, pitch and roll. Is it necessary to have mathematical model for rov/auv? how do i determine the model? Please can you also teach me whether kalman filter is necessary or not, if yes then how is it implemented, what are the steps, inputs and outputs? i hope you can help

22. asdfafadsf says:

The low pass filter formula is wrong. It must “+” rather then “-“.

1. AnilM3 says:

Fixed

23. Solmer says:

Hi ! I’m using a seeeduino Mega 2560 + GY-87 module.I’ve attached the moudele to the 2560 board (GND to GND, VCC_in to +5V, SCL to pin ANALOG 5 and SCA to pin ANALOG 4) and I’ve compiled the code but when i opne the monitor the values don’t change, I get the same value all the time….

Any ideas?

Thanks in advance for your help.

24. sls says:

It is interesting to know accuracy level of the angle measurement achieved using this set up?

Is it only static measurements. Hopefully good enough for slow varying angle(less than 20 cycles per sec.) measurements.

25. Shweta says:

Hi Anil,
This blog was of great help, thank you. You did talk about your next blog about gyro and its intergration with acc where can i find it ? I am facing some issues when i intergrate acc and gyro. It would be of great help if i could get some inputs from you about gyro. Thanks in advance.
Regards,
Shweta

1. I’m also looking for similar information, if you find anything please let me know! 🙂

26. baicker says:

Very good for a newer, thanks very much!
But could you explain the formula from arctan to atan2 more detail ?

27. mohsein says:

Hello

thank you for the nice effort

how can we make visualization like what you did

tell us about that pleas

thank you again

28. Nice article, I needed something simple and clear like this, thank you 🙂

29. Allmin says:

Hi,
I would like to know how Yaw can be calculated

30. Thanks for the insight! Especially how you create the low pass filter I find transparant enough to look into and hope to understand shortly.

31. Matthew Judijanto says:

Can I ask for the gyroscope Fritzing file? Currently I’m making a project with the same gyroscope and I need that fritzing drawing

32. Mahim says:

Hi,.,
I am working on an Arduino based IMU module which will be used to determine human joint angles. I was going through articles and came across your tutorial page and found it extremely helpful to understand the basics. Thank you!!

The Inertial sensor I am using is the GY-80 module (I am using only accelerometer ADXL345 and Gyroscope L3G4200D)

When comparing the Accelerometer angle data with that of the gyroscope, I find that they are not really similiar. (The code I am running is based on several codes found online).

For the accelerometer angles I have used the formulae:

angle_x= (atan2(xg, sqrt(yg*yg + zg*zg))*57.2957795);
angle_y= (atan2(yg, sqrt(xg*xg + zg*zg))*57.2957795);
angle_z= (atan2(sqrt(xg*xg + yg*yg), zg)*57.2957795);

And for angles using gyroscope, I have integrated them using the trapezoid method (I hope I am calling it right!!!)

I placed a micros() timer at the end of the setup loop and have reset the timer at the end of the angle computation part.

angle_gx += gx*((double)(micros()-time1)/1000000);

angle_gy += gy*((double)(micros()-time1)/1000000);

angle_gz += gz*((double)(micros()-time1)/1000000);

I have spent a lot of time in this and I am not able to proceed further 😦

I would be highly grateful if someone could help me with it!!! Are the formulae mentioned above correct???

Please let me know if I should copy the code and results as well!1

Thank you

33. Hello,

I was looking into your formulas and the code that you implemented. On the formula for pitch the numerator is Gy while on the code you are using fXg and this isn’t the only deviation from the formulas since there are many.

Which one is correct? The formula or the code?

1. AnilM3 says:

Depends on how you position your accelerometer and which rotational axis is going to be the roll of your system.

1. Could you please elaborate this. I thought that the rotation axis always was defined as Roll = x-rotatio, Pitch = y- rotation and Yaw = z-rotation. Also how does the position of the accelerometer influence the formular?

34. Zershan says:

Hi
Can i use gy-85 or gy-50 with same code?both have same chips you mentioned

1. AnilM3 says:

If they have the same chip and they’re using the same bus, I don’t see a reason for it not to work. In any case, I haven’t tested it myself so I can’t guarantee that it will.

1. Zeeshan says:

Actually I used gy521 and hmc5883l as separate modules.while using anyof these at ine time the values are not good….like gyro x,y,z continously increment even if gy521 is stable flat on table…any idea?

1. AnilM3 says:

I never mentioned the gyro in this particular post, but cheap gyros are prone to drift.

35. Khang Nguyen says:

“Ata2(-fYg, fZg)*180.0)/M_PI” quote
What is M_PI?

1. AnilM3 says:

Number Pi

36. Alejandro Cabezas says:

Hello! I would like to know, when you do the convertion to Gaccel, the Raw accel is a digital entry? or is the analog output from the accelerometer. Thank you

37. Hyon Kim says:

Hi Anil

Thank for the great summary of the topic.
I have a quick question; Do you have performance comparison of roll and pitch directly received from IMU sensor vs roll and pitch derived from the formula after low pass filtering?

I appreciate if I could see that too.

Thank you so much.
HK

38. Hai, my mane is hakim. I am electrical engginering student from indonesia. Now, i’ve a research in Indonesian institut of science about IMU. It’s really confusing to understand how to process signal using kalman filter. Can you make a posting about kalman filter?. I really need an algorithm about kalman filter.

39. I’m using unity with a LIS2DH accelerometer..

Because pitch/roll/yaw only make sense given a fixed point of reference this is causing all kinds of strange artifacts like drifting, gimbal lock and cross-contamination on the axis that I didn’t ask to change!

I’ve read some people have had luck creating an artificial horizon.. they project a fixed vector and a variable one onto a common X/Z plane, then they apply the yaw ,for example, around the global axis instead of the object’s axis. Although I’ve read even this isn’t perfect, and it’s better to use local rotations to update the forward vector together with a reference up vector to build the final orientation.

Unfortunately, I haven’t seen an actual implementation for any of this… I understand only 2 angles (pitch,yaw, or roll) can be derived from a 3-axis acclerometer, but what does the unity code look like transform.rotate,LookRotation,ProjectOnPlane, Quaternion.Euler, etc?

40. dharm says:

I need to calculate distance when we have accelerometer and gyro readings
it mean we have acceleration in xyz and gyro in xyz so how we can calculate distance please

41. Hi,
In your code
roll = (atan2(-fYg, fZg)*180.0)/M_PI;
pitch = (atan2(fXg, sqrt(fYg*fYg + fZg*fZg))*180.0)/M_PI;
But According to Eqn 25 and 26 in AN3461 , the negative sign should be applied to pitch , not for roll ?

Code should be
roll = (atan2(fYg, fZg)*180.0)/M_PI;
pitch = (atan2(-fXg, sqrt(fYg*fYg + fZg*fZg))*180.0)/M_PI;