Let's start with Atlanta, a function that I have mentioned in the last lesson Atlanta function is one
of the most important functions during programming, and you will see its applications in robotics quite
frequently.
Its purpose is the same with architectures or inverse tangent function, namely finding and from the
value that has been obtained by tangent function.
Let's be more clear.
Imagine a right angle triangle with sides of A and B and Angle Tito.
As we know from Trigonometry Tangent Tito is given by this formula.
We can find data from this formula by finding inverse of it, namely inverse tangent.
While it's very simple this method, Holmes has some serious problems.
Let's assume that we have a circle that represents all the angles we can have from zero to two PI.
Let's try to calculate tangent of PI over four, which is nothing but division of Sun Pi over four and
Cosine Pi or four, which will make one.
Now let's take three pi or four radians and calculate its own tangent.
You will see that its tangent is also one.
So as you can see, we lost information here.
Namely, one can be result of inverse tangent of PI or four PI or four and three PI or four, and with
simple arc tangent function, we cannot know that.
So the range of upon function or a tangent function is minus PI or two and plus PI or two exclusive
exclusive because cosine plus or minus PI over two is zero, which causes discontinuities, which is
another problem of our tangent function.
However, with a ton to function or acting is to function.
We don't have any of these problems.
Its range is zero two to a point, which is whole circle, and there is no discontinuity at plus or
minus PI over two.
If you think carefully, the reason that Atom two forks is that in the case of atom function or Klingons
function by division, we lost sine information because two sons signs from sine and cosine functions
collapse into one sine either a plus or minus.
That creates uncertainty, which is not the case in order to function because we don't divide.
We give two inputs and let a time to do the rest of the work.
OK, it's not about Atlanta function, let's continue to our last topic.
Now we will see another two methods of representing rotation.
Let's start with axis angle representation.
Rotations are not always how to be informed about the principle axis, namely X, Y and Z.
Indeed, we can represent in the rotation by a single rotation around some axis by a specific angle
of Keita.
We can find this rotation axis by analyzing eigenvalues and eigenvectors over rotation matrix.
Let's take this rotation matrix as an example and try to find eigenvalues and eigenvectors of it.
If you analyze its eigenvalues, you will see eigenvalue with real value of one and its corresponding
eigenvectors from eigenvalue and eigenvectors relation.
We know that matrix times eigen vector equals the eigenvalue times eigenvectors.
In our case, eigenvalue is one, so formula becomes like that.
This means that we multiply a vector with a matrix and we don't get any change in its magnitude.
So this matrix can be on a rotation matrix because rotation matrix only cause rotation, no change in
the absolute value of the vector or length of the vector.
So I can vector that corresponds to the eigenvalue of one is the axis of rotation.
We can find the rotation angle by comparing eigenvalues with this formula.
Don't worry, we will see this clearly in our next lesson, which will be practical applications of
these.
So this is our angle and rotation axis that represents observer rotation matrix.
However, this answer is not unique.
Namely, the rotation of data around minus v axis results the same rotation.
So this angle?
This is the representation we have for quantities to represent a rotation, namely one quantity for
representing the rotation angle and three quantities for representing the rotation axis.
However, not all of these are independent as our rotation axis is a unit vector.
Two of its values Unit Vector two of its values are independent.
The third one is defined based on the other two, so we have three independent quantities.
We can describe axis angle representation by this formula.
TITA is the rotation angle, and the key is our rotation axis.
As Kate is a unit vector, the model of Vector R will be angle tita to and its direction will give rotation
axis of K.
However, it's important to note that we cannot do a vector algebra or vector are named.
We cannot combine to axis angular representations using standard rules of vector algebra because this
would mean that rotations compute, which is not correct.
Let's continue with unit quarter news, which is very important, and once the user representation of
rotations in robotics quaternary in our extension of the complex number name, the hyper complex number,
and it can be written as a scalar plus a vector parts.
So this represents the scalar part and V represents the vector part.
I would like to mention you also that I, J K are orthogonal complex numbers and they're square and
their multiplication with each other gives negative one.
We can additionally represent Kryptonians in this way.
Quote and support addition and subtraction operations, which is performed Element Y.
As you can see below, the example of addition of two quarter new addition is associative and commutative.
Apart from the addition and subtraction, Cortana's also support continuum, or Hamiltonian product.
This represents a relative rotation, as in the case of rotation mattresses.
Here is the Hamiltonian product formula as in the case of rotation mattresses, post multiplication
gives rotation with respect to the current frame, while print multiplication give rotation with respect
to the fixed frame to represent rotations.
We use unit quarter news unit continues our subcategory of continuous and they represent rotation of
T to about a unit axis of V.
We generally refer to them as simply quaternary news.
They are represented in this way.
From this form love, we can get unit quarter in use that represents rotation around principle axis.
For unit quarter use, their magnitude is equal to one.
So why do you turn use instead of roll PTO or a lower angle representations?
Even if the visualization of unit quaternary use is complex because they are compact?
They are computational efficient.
Also the most important advantage that they don't have Gimbel look that Aler and.
All pitch your angle representations are prone to.
We will see this problem in the next lesson.