Fixed and Euler Angle Representation for Rotation Matrices


hi everyone
in this video we will be looking at fixed and Euler angle representations
for rotation matrices if you aren’t quite familiar on how rotation matrices
work check out my previous video on spatial descriptions and transformations
by clicking the link up here right so now we have two main ways in which we
can represent a rotation of one frame with respect to another the first being
the fixed angle notation so here if you have two frames the frame in red we can
call it B and this would be the moving frame on the frame that we are going to
rotate the frame black you can call it a and this would be the fixed or the
reference frame so it undergoes no rotation in a fixed angle notation when
a rotation is performed it is rotated about the fixed or the reference frame
so in this case if you want to rotate B about the x axis it would be about A’s x
axis by an angle theta and this would look something like this where you have
frame B rotated about A’s x axis this can be represented in this kind of
equation so here we have done an X rotation followed by a Y and then a Z
rotation however when calculating this you would
work backwards by calculating the Z rotation first then the Y and finally
the X rotation the second kind of rotation would be an Euler angle
representation so here we have our two frames frame B which is the moving frame
and frame A which is the fixed frame for the Euler angle representation a
rotation would take place about the moving frames own axis so if we wanted
to rotate frame B about the Xx axis it would be about its own
axis by an angle theta and this would look something similar to this this can
also be represented using a similar equation so here we have done a rotation
along X followed by Y and then Z the notation X prime is used to show that it
is a Euler angle representation when calculating it you would do it in the
same order as you rotated it so X then rotation around Y and finally a rotation
around Z so in summary what you have to know is that both fixed and Euler angle
notations give the same final orientation on the same final rotation
matrix what this means is that for your fixed angle notation you had rotation of
X followed by Y and then Z you would calculate it backwards and so you would
get your Z rotation multiplied by your or Y and your X this is because you’re
rotating it about a fixed or reference frame you could that you calculated
backwards this would give you the same answer if you use two Euler angle
notation we have a rotation of XYZ you would calculate it in the order of
rotation so you would get your rotation of X multiplied by Y multiplied by Z and
this is because the moving frame is rotated about its own set of axes and so
it is done in the order that it is rotated now there are multiple types of
fixed and Euler angle notations such as Z Y Z or Z Y X for example there are 24
of these sets and together they form the angle set convention there are also
other types of notations used to represent a rotation such as the
equivalent angle axis representation but I will not cover that in this video
all right guys so that’s all for this video if you guys liked it please click
the like button below if you have any questions or clarifications on fixed and
anger representations please feel free to leave a comment and I will do my best
to get back to you I hope this video helped and if you haven’t done so
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