Will,

In answer to your question, I am sorry, if you want to run the DCM algorithm, you will need three gyros. There is no way to get around it.

If you want to get away with fewer gyros, you can do that, but you will need to use a different algorithm.

When I started out in this hobby, I was using a board that had a yaw gyro and a pitch gyro, without a roll gyro. It also had a GPS and a 2 axis accelerometer. I was able to achieve control and navigation of a Gentle Lady without any problem, using complementary filter.

I think it is going to be difficult for you to control yaw without a yaw gyro, using GPS alone, unless you use a GPS with a high update rate, such as a uBlox. The problem with a GPS with a 1 Hz update rate is that the control yaw control loop will go unstable if you turn the gain up, and the wind will blow your plane away if the gain is too low.

However, the situation you propose is different from the one that I tried, since you are proposing a roll gyro. Maybe it would work. But if I were using only two gyros, I would use yaw and pitch.

You know, you could buy the UAV DevBoard for $150, a PICkit2 programmer from Microchip for $40, and an EM406A GPS from SparkFun for $60. That is only $60 over your budget.

Best regards,
Bill

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Here is Will's question:

Thankyou so much for your help that really clears things up for me. I have just one other question for you. Is it possible to implement the DCM on only the Pitch and Yaw axis? The reason I ask is that one of my goals in developing this device is to keep the cost below $200. I felt that I could achieve satisfactory turns with only pitch and roll data coming from the IMU and the Yaw being updated once a second from the GPS. The extra gyro is going to blow my budget :) What are your thoughts on this.

Will,

Jordi is using the DCM algorithm, based on Robert Mahoney's ideas.

The DCM algorithm maintains the 9 elements of the direction cosine matrix that represent the rotational orientation of the plane with respect to the ground. The basic idea of the method is:

1. Perform direct numerical integration of the nonlinear differential equations that relate the time rate of change of the 9 direction cosines to the 9 direction cosines and the roll, pitch, and yaw rates measured by gyros in the frame of reference of the plane.

2. Adjust the 9 elements of the matrix to enforce the orthogonality relationships that the rows and columns of the matrix must satisfy.

3. Use nonlinear complementary filters to compensate for gyro drift using accelerometer and GPS information.

There are no singularities in the computations in the DCM algorithm.

You can express all control and navigation computations in terms of the 9 direction cosines, but some folks prefer working in terms of Euler angles, so Jordi computes Euler angles from the direction cosines.

Regarding the difference between the DCM algorithm and a linear complementary filter, the DCM algorithm is much more accurate in all orientations, since it explicitly represents the nonlinearities of the rotation group at all stages of the computations.

For the theory of the DCM algorithm, take a look at Mahoney's papers or this summary.

Best regards,
Bill Premerlani