what you are looking at right now are all possible unique sequences of euler
angles which can be used to describe the
orientation of an object in three dimensional space
although if you think about it the euler angles are rather defined in an
asymmetric manner which looks a little confusing at first
after watching this video you will have some intuition for their definition
and see how useful they are in describing rotations in 3d
first i will explain the necessity of euler angles why do we care about it
then i will explain how euler angles helps us to describe the orientation of
an object in space let us start with rotations in
2d if a particle rotates with angular
velocity omega about a fixed axis we say that omega is the rate of change
of the angle theta where theta is the angle through which
the body has turned about that fixed axis
we use this angle as a coordinate to describe the position of the particle in
2d in three dimensions the angular velocity
vector has three components but there is something strange about
this vector the three components of omega have no corresponding angles
to serve as coordinates unlike in 2d now there exists a possibility of
different axis of rotation and we no longer have such natural
angles to define the orientation of a rigid body
also in 3d finite angular rotations are not vectors
since they do not commute and we do not have three angles which can be used as
coordinates we are therefore compelled by the nature
of angular rotations to look for some set of three angles
to describe the orientation of a body the usual set of such angles are called
euler angles which although being unsymmetrical are
very useful in describing their rotational motions in three dimensions
any general motion of a rigid body can be represented as a translation
plus a rotation and if one point on the body is fixed only
rotational motions are possible we will only pay attention to such
movements or intrinsic rotations of an object
as they are relevant to the notion of euler angle
the key ingredient here is a theorem by Euler himself
which guarantees that any motion of a rigid body which has one fixed
point is equal to a rotation about an axis passing through that fixed point
what this means is that for every such general rotation
we can find an axis through the fixed point with some particular alignment
such that a rotation by a precise amount about that axis will give us that exact
general rotation
to describe such intrinsic rotations of the rigid body
it is conducive to introduce two coordinate systems one always fixed in
space and one that is attached to rigid body
which will participate in each rotation of the body
the relative orientation of these two orthogonal systems will be described by
the euler angles this is equivalent to saying that using
euler angles or three rotations we can go from one frame to the other or
any rotation can be decomposed as a product of three elementary rotation
matrices the three euler angles are typically
denoted by phi theta psi or alpha beta gamma for the
sake of this video i will call them phi theta and psi to
see how it all works suppose the rigid body is being rotated
into some arbitrary configuration with respect to a fixed coordinate axis
we want to define this orientation through a sequence of rotations
starting from the initial orientation of the body fixed frame
let us call this the frame 1 the space fixed frame
the frame 2 the initial body fixed frame and the frame 3
the final body fixed frame
there are plenty of ways we can rotate the frame two with respect to frame one
we can rotate the frame to about the x y z axis of the frame one
or x y z axis of the frame two by an angle
phi theta or psi this gives us a total of 3 times 3 times 3 equal to 27
sequences of Euler angles however two consecutive rotations about
the same axis are redundant as we can get to the same
result by rotating around the same axis twice what we can have are rotations
like x y x etc since a rotation about y
after x changes the original x axis so the next time iteration about x will
be around the different or new x-axis we just get a total of 3
times 2 times 2 equal to 12 unique possible sequences of
rotation to get to the final configuration
or frame 3 from frame 2. the pedagogical approach to euler angle is the analysis
of the heavy symmetric top with one fixed point
you can now recognize the rotation sequence that was used in this analysis
it is a z x z sequence the first rotation is phi
about the z axis the rate of change of which gives us the precision of the top
the second rotation theta is about the x-axis the rate of change of this angle
is called nutation
and the third rotation psi is about the z axis
which corresponds to the intrinsic rotation of the spinning top
remember how we started with the problem to set up angles for the three
dimensional rotating bodies let us now express the angular velocity
omega along the x1 x2 and x3 in terms of the three euler angles and their
derivatives the angular velocity has three
components omega 1 along x1 omega 2 along x2 and omega 3
along x3 each of these components of omega
has contribution from the rate of change of each of the euler angles
let us first evaluate the rate of change of the three euler angles along the
three moving axes
consider the end state of two axis after a zx z
sequence of euler rotation using the right hand screw rule we can find the
direction of theta dot phi dot and psi dot the vector theta dot
has the components theta 1 along x 1 x theta 2 along x 2 and theta 3 along x 3.
it makes an angle psi with the x 1 axis in the clockwise direction
and 360 degree minus psi in the clockwise direction
resolving theta dot vector along x1 and x2 and
x3 we can find the three components of theta dot
the angular velocity phi dot makes an angle theta with x3
and its projection on the x1 x2 plane makes an angle psi with the x2 axis
the component of phi dot along x3 is phi dot cosine of theta
and the projection on the x1 x2 plane is phi dot sine of theta
resolving this component further in the x1 x2 plane
we can get the components of phi dot
finally the angular velocity psi dot is only along the x3 axis
collecting all the components we get the complete expression for the angular
velocity vector of the rigid body with this we have solved our problem of
representing the orientation of an object in 3d
i hope you have understood why the euler angles are defined the way they are
i'll see you in the next video