Direct Cosine Matrix
Direction Cosine Matrix IMU: Theory
William Premerlani and Paul Bizard This is the first of a pair of papers on the theory and implementation of a direction-cosine-matrix (DCM) based inertial measurement unit for application in model planes and helicopters. Actually, at this point, it is still a draft, there is still a lot more work to be done. Several reviewers, especially Louis LeGrand and UFO-man, have made good suggestions on additions and revisions that we should make and prepared some figures that we have not included yet. We will eventually incorporate their suggestions, but it may take a long time to get there. In the meantime, we think there is an audience who can benefit from what we have so far. The motivation for DCM was to take the next step in stabilization and control functions from an inherently stable aircraft with elevator and rudder control, to an aerobatic aircraft with ailerons and elevator. One of the authors (Premerlani) built a two axes board several years ago, and developed rudimentary firmware to provide stabilization and return-to-launch (RTL) functions for a Gentle Lady sailplane. The firmware worked well enough, and the author came to rely on the RTL feature, but it never seemed to work as well as the author would like. In particular, satisfactory solutions to the following two issues were never found: • Mixing. It was recognized that in a banked turn, there were two problems arising from the bank angle. First, the yaw rotation of the aircraft around the turn generated a nuisance signal in the pitch gyro, because of the banking. Second, in order to make a level turn, the elevator needed some “up” deflection. The amount of deflection depends on the bank angle, which could not be directly measured. Both issues were opposite sides of the same coin. Acceleration. An accelerometer measures gravity minus acceleration. The acceleration is equal to the total of all of the aerodynamic forces (lift, thrust, drag, etc.) on the plane, plus the
William Premerlani and Paul Bizard This is the first of a pair of papers on the theory and implementation of a direction-cosine-matrix (DCM) based inertial measurement unit for application in model planes and helicopters. Actually, at this point, it is still a draft, there is still a lot more work to be done. Several reviewers, especially Louis LeGrand and UFO-man, have made good suggestions on additions and revisions that we should make and prepared some figures that we have not included yet. We will eventually incorporate their suggestions, but it may take a long time to get there. In the meantime, we think there is an audience who can benefit from what we have so far. The motivation for DCM was to take the next step in stabilization and control functions from an inherently stable aircraft with elevator and rudder control, to an aerobatic aircraft with ailerons and elevator. One of the authors (Premerlani) built a two axes board several years ago, and developed rudimentary firmware to provide stabilization and return-to-launch (RTL) functions for a Gentle Lady sailplane. The firmware worked well enough, and the author came to rely on the RTL feature, but it never seemed to work as well as the author would like. In particular, satisfactory solutions to the following two issues were never found: • Mixing. It was recognized that in a banked turn, there were two problems arising from the bank angle. First, the yaw rotation of the aircraft around the turn generated a nuisance signal in the pitch gyro, because of the banking. Second, in order to make a level turn, the elevator needed some “up” deflection. The amount of deflection depends on the bank angle, which could not be directly measured. Both issues were opposite sides of the same coin. Acceleration. An accelerometer measures gravity minus acceleration. The acceleration is equal to the total of all of the aerodynamic forces (lift, thrust, drag, etc.) on the plane, plus the
References: Several papers written by Robert Mahony et al are available at http://gentlenav.googlecode.com/files/MahonyPapers.zip. DCM 30 Draft: 5/17/2009