Quaternions vs atan2

[nyd]

Hi!

First of all, congratulations for your project.

I'm developing a movement quantification system, with a 3-axis gyro and a 3-axis accelerometer.
In my first approach i'm only using the 3-axis acc to get the rotation using for that purpose the atan2 function and the acceleration vectors.

Two questions, did you test this approach? And a newbie one, how did you used the quaternions with the gyroscopes?

thanks in advance for your answer.

Regards

[Mitch]

I'll jump in,

Using the inverse trig functions is a very common solution but there are several drawbacks. They have singularities at specific attitudes that have to be addressed separately. They converge at the north and south poles where roll is the same as yaw (where pitch = pi/2). The computation time for the trig functions is higher than for the simple multiplication required for manipulating quaternions. A direction cosine matrix is 9 scalar numbers where a quaternion rotation is only 4.

Here is an example of a routine that estimates the attitude from accelerometer readings:

/*
* This will convert from accelerometer and compass readings into
* [phi, theta, psi] (in radians).
*/
void accel2euler(
vector euler,
float ax,
float ay,
float az,
float compass
)
{
float g = sqrt( ax*ax + ay*ay + az*az );

euler[0] = atan2( ay, -az ); // Roll
euler[1] = asin( ax / -g ); // Pitch
euler[2] = compass; // Yaw
}

Note the accelerometer data alone is insufficient to determine the attitude because the accelerometers cannot measure the rotation about the gravity vector (vertical axis). Some other data is required. The g vector magnitude calculation is not required but it helps to minimize error due to accelerometer calibration. Any acceleration of the vehicle will be measured by the accelerometers and render the attitude estimate unreliable.

Quaternions eliminate the difficiencies I listed above. Accelerometers must be augmented with other data.

In Tom's code, he elegantly implemented a control loop that rotates the quaternion estimated attitude based on gyro measurements.

[Tom]

Mitch already formulated a nice answer

I would propose to take a look at the code and do some tests or simulations.

[nyd]

In did a good answer.

One question, in theta, why can't you calculate it using the atan2 function, like this:

theta= athan2(ax,az) ? similar for the phi rotattion.


From what i understood, since you are unable to get the vertical rotation using the accelerometer data, to develop a 3-axis rotation system you always need to have a 6-DOF data.

Whell... at least a 4-DOF system.

Thanks for your answer, once again.

[Tom]

It depends on how you want to have your angles.
For example:
pitch = 110°, roll = 0°, yaw = 0°
pitch = 70°, roll = 180°, yaw = 180°
These 2 attitudes are the same. I'm sure you can come up with some less obvious examples
In my code, pitch never exceeds 90°, while roll does. Which means an asin would be fine for pitch.

The needed tools depend on what you want. You could have a 2 axis accelerometer and 1 yaw gyro without drift.
You could also have 3 gyros, 3 accelerometers and 3 magnetometers. A 9-DOF that does the same thing
But that's all theory. 6DOF defines that we have a system with 6 inputs which we use in our model.

[nyd]

Hmmmm... good point.

I searched for you implementation of quaternions but didn't found it on the download section. In which .c is it?


Regards.

[Tom]

I really should make the sourcecode more viewable somehow

Here you can find the quaternion implementation: http://code.google.com/p/gluonpilot/sou ... aternion.c

[nyd]

Hi Again.

I was studying your code and in the update of the quaternion with the input rates you have:

Code: Select all

  q0 += 0.5f * (        - q[1]*w1 - q[2]*w2 - q[3]*w3)*dt;
        q1 += 0.5f * (q[0]*w1 +           q[2]*w3 - q[3]*w2)*dt;
        q2 += 0.5f * (q[0]*w2 - q[1]*w3 +           q[3]*w1)*dt;
        q3 += 0.5f * (q[0]*w3 + q[1]*w2 - q[2]*w1)*dt;


But here
http://www.euclideanspace.com/physics/kinematics/angularvelocity/

they present this differential equation:

d q0(t) / dt = − 1/2* (Wx (t) q1(t) + Wy (t) q2(t) + Wz (t) q3(t))
d q1(t) / dt = 1/2* (Wx (t) q0(t) + Wy (t) q3(t) − Wz (t) q2(t))
d q2(t) / dt = 1/2* (Wy (t) q0(t) + Wz (t) q1(t) − Wx (t) q3(t))
d q3(t) / dt = 1/2* (Wz (t) q0(t) + Wx (t) q2(t) − Wy (t) q1(t))

The difference between both case is the negative sinal when updating (for example) q2:

Code: Select all

+ Wz (t) q1(t) − Wx (t) q3(t)

and

Code: Select all

 - q[1]*w3 + q[3]*w1

is this an error ?

[nyd]

I searched for an answer and your code is the one thats right.

Regards.

[nyd]

Another step another doubt

Do you combine the Gyros data (rates) with accelerometers in the quaternions? and How?

Or you just do that in the Kalman filter.

Regards.