euler angles 123

7/25/2019 Euler Angles 123

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xe

ye

ze

xb

yb

zb

Axis conventions

To describe the motion of an airplane it is necessary to define a suitable

coordinate system. For most problems dealing with aircraft motion, two

coordinate systems are used. One coordinate system is fixed to the earth andmay be considered for the purpose of aircraft motion analysis to be an inertial

coordinate system. The other coordinate system is fixed to the airplane and is

referred to as a body coordinate system. Figure 1 shows the two right-handed

coordinate systems.

Figure 1 Body fixed frame and earth fixed frame

The orientation of the airplane is often described by three consecutive

rotations, whose order is important. The angular rotations are called the uler

angles. The orientation of the body frame with respect to the fixed earth frame

can be determined in the following manner. !magine the airplane to be

positioned so that the body axis system is parallel to the fixed frame and then

apply the following rotations"

1. #otate the body about its $b axis through the yaw angle

%. #otate the body about its yb axis through the pitch angle

&. #otate the body about its xb axis through the roll angle

1


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yb

xb

zb

Figure % Body axes coordinate system

Direction cosine matrices

'ertain types of vectors, such as directions, velocities, accelerations, and

translations, (movements) can be transformed between rotated reference

frames with a &*& matrix. +e are interested in the plane frame of reference

and the ground frame of reference. !t is possible to rotate vectors by

multiplying them by a matrix of direction cosines"

Qx

Q=

Qy

=a vector, such as a direction, velocity or acceleration

Qz

QP=a vector Qmeasured in the frame of reference of the plane

QG=a vector Qmeasured in the frame of reference of the ground n. 1

rxx rxy rxz

R=

ryx ryy ryz

= rotation matrix

rzx rzy rzz

QG= RQP

The relation between the direction cosine matrix and uler angles is"

R =

coscos

cossin

sinsincos cossin

sinsinsin+ coscos

sincos

cossincos+ sinsin

cossinsinsincos

coscos

n. %

2


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sin

uation 1 and euation % express how to rotate a vector measured in the

frame of reference of the plane to the frame of reference of the ground.uation 1 is expressed in terms of direction cosines. uation % is

expressed in terms of uler angles.

!f we have the full direction cosine matrix, we can convert to uler angles

from the last row and the first column of the matrix" = arcsin(rzx)

= atan2(rzy,rzz) n. &

= atan2(ryx,rxx)The pitch angle is between - degrees and / degrees. 0ote that we

must use atan% in order to get a four uadrant result. Finally, note that the

atan% function taes its arguments as (y,x), not (x,y). That often leads to

confusion.

2ome people have run into problems in trying to apply euation & to

computing uler angles using the direction cosines that are computed by the

firmware, such as 3atrix4ilot, that runs on the 567 8evBoard.

The main problem is that the 567 8evBoard does not use the coordinate

system shown in figures 1 and %. !nstead, the 567 8evBoard uses yb where

xb is in Figure %, and it uses 9xb where yb is in Figure %. The reason for that

is historical" the axis labels were placed on the board to align with the axes of

the three axis accelerometer chip. :ater, we discovered that was not theconvention, but by then it was too late. The relationship between the direction

cosines used in the 567 8evBoard, and the ones given in uation 1 and

uation % is"

rmat[0] rmat[1] rmat[2] ryy ryx ryz

R=

rmat[3]rmat[4]rmat[5]

=

rxyrxx rxz n. ;

rmat[6] rmat[] rmat[!] rzy rzx rzz

2o, in terms of the elements of the array rmat

euler angles 123

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