To express the orientation of an object in the three dimensional space.
So far we have used rotation matrices.
With this tool, we are able to simplify and decompose the problem of orienting an object in the three
dimensional space around three axes with three elementary rotations.
With this tool, we are able to simplify and decompose the problem of orienting an object in the three
dimensional space around three axes with three elementary rotations.
A body can rotate around the Z axis.
So if we call this rotation angle of the robot around the Z axis theta, then the orientation of the
robot is defined by an elementary rotation matrix that we have calculated, and that is uniquely a function
of the angle theta.
Similarly, it can also rotate around the y axis this time, the angle that express the rotation of
the object around the Y axis is called Phi.
And once again, we can write an elementary rotation matrix that express the orientation of the robot
with the respect to the y axis only.
And finally, an object can also rotate around its x axis by an angle PC.
The orientation of a robot around this axis can also be expressed with a new elementary rotation matrix
that is only a function of the angle PC.
If we consider the general case of a robot or generally of any object that can freely rotate and orient
itself in the three dimensional space, then the representation of the orientation in Euler angles tells
us that we can decompose the orientation of an object into three elementary components.
Three elementary rotations around the X, Y, and Z axis.
If we call theta the rotation angle around the Z axis Phi, the rotation angle around the Y axis and
C the one around the X axis.
For each of these elementary rotations, we know how to write the respective rotation matrix and therefore
the overall orientation of an object is given by the composition.
And so the product of these three elementary rotation matrices, these three elementary components of
the orientation of an object in space are commonly known as roll pitch.
And you.
By multiplying these three matrices, we obtain a new matrix whose components will depend on the three
angles.
So Theta Phi and PSI.
In conclusion, with this convention of Euler angles to express the rotation of an object in the three
dimensional space, we need nine components.
That is, to express the orientation of an object with rotation matrices.
We need to calculate and store the nine components that form the rotation matrix.
Furthermore, if we want to compose successive rotations, that is, if we want to multiply rotation
matrices, we will have to perform a matrix product between two matrices with three rows and three columns.
And this mean calculating 27 products and 18 sums.