Let's not discuss some of the key properties of rotation mattresses and their composition before going
further, let me explain something, as we said before viewpoint rotation matrix by finding the projection
of axis or each other, which is nothing but dot product.
So if you want to find the rotation matters between two frames, you can easily do that by just finding
dot product between the axis of these frames.
So let's see an example assume that we have two frames, namely oh zero and or one or one has been obtained
by rotating or zero by two degrees about zero axis.
Then we can write R1 zero like that before continuing on.
Let me mention the formula of DOT product here.
So Dot product of two vectors is nothing but products of the lengths of two vectors and cosine of angle
between them.
Let's copy the graph and rotation matrix here again.
Here you can see the calculation of each entry of the rotation matrix.
Don't forget that in rotation mattresses, we are handling unit vectors, namely the lengths of vectors
this one.
So we will only have the cosine of angle between them.
Moreover, the dot product of orthogonal vectors will be zero because the angle between them is PI over
two and cosine of PI over two is zero.
Additionally, axes that are parallel and are on the same direction will have an angle of zero between
them, so their product will be one because cosine of zero equals to one.
That's right, the resultant rotation matrix.
As you can see, we easily got the elemental rotation matrix, which is obtained by rotation among x
axis about x axis.
Don't forget to take into account that during the calculations of dot products, you are handling three
dimensional case here.
Additionally, I would like to note something else.
Moreover, you can see it from rotation metrics that each row contains the direction.
Cosine is a one axis of frame over zero with respect to the axis of frame or one direction cosine.
Because we have dot product of unit vectors, namely cosine of angle between them, as we mentioned
earlier.
Also, you can see that each column of rotation matrix contains the direction cosine of one axis of
frame or one.
With respect to the axis of frame zero.
Yeah.
If you have taken robotics course, probably you have seen these definitions and I hope that, you know,
understand what they mean.
Let's go to the properties of rotation mattresses, rotation mattresses in three.
They belong to the spatial or technical group of Order three, which have the below properties or transpose
equals that are inverse.
The column and therefore rows of are mutually orthogonal.
Each column and therefore each row of our is a unit vector determinant of R equals to one, which means
that the length of a vector is unchanged after transformation, which is very initiative because what
we are doing is just rotating the frame rotation.
Matrix has nine elements in three D case, but they are not independent from second and third properties
of the rotation mattresses.
We get six constraints, namely, the columns have unit magnitude, which provides three constraints.
The columns are orthogonal to each other, which provides another three constraints.
So nine elements with six constraints means that we have three independent value.
This means that the rotation mattresses are not the most compact representation of the rotation because
we have dependent values.
That is why in future releases, we will see more compact representations or rotations.
Now we can talk about the composition of rotation mattresses.
Keep in mind that generally we have more than one rotation of frames with respect to each other.
We can generalize these rotations into sections first rotation with respect to the current frame.
Second, rotations with respect to the fixed frame, let's go with rotation, with respect to the current
frame.
What's the current frame?
Let's assume that we have a frame of zero.
Then we rotated a degrees around axis of x zero and get frame of or one before the current frame was
always zero.
However, after rotation of T to the current frame is no or.
Now we rotate one more time, better degree about the y one axis of current frame and we get the new
current frame of all two.
I think this is enough for understanding.
No, the problem is how we can combine these rotations.
For example, we want to get the rotation matrix of R to zero, which represents rotation from frame
two to zero.
Let's assume that we have a point p and its representation on these three frames are P zero, p one
and P two.
Then based on simple rotation matrix formula that we have.
Seen before we can write these equations.
So by combining equations one and two, we can write this equation by comparing equations four with
equations three, we can see that are two zero equals two product of our one zero and our two one.
So when the subsequent rotations are about the axis of current frame, we just past the post multiply
rotation mattresses.
Let's now see.
The key is when the subsequent rotations are about the fixed frame.
So assume that we have a frame of zero v rotate this frame by two degrees around zero and get new frame
of one.
Then we once again rotate the frame of all one of otherwise zero axis of frame zero better degree.
As you can see, the rotations are about oh zero, namely fixed frame.
So what is the overall rotation from frame two to zero in this case?
In this case, we will multiply the rotation mattresses to get the final rotation matrix of our two
zero.
So as you can see if subsequent rotations are about the axis of fixed frame, then we print multiply
the rotation mattresses.
Let's call this section by a simple example we have below rotations first rotation of data about the
current x axis.
Second, rotation of fear about the current that axis served rotation of alpha above the fix that axis
for rotation of better about the current y axis and fifth rotation of tilt to fix the x axis.
Don't forget the composition rules for current and fixed frames.
So in order to determine the cumulative effect of these rotations, we begin with the first rotation
R s T to and pre or post multiplied, depending on whether we rotate about the current or fixed frame.
So let's right step by step.
First, we have the rotation of our data, then the post multiply R and R Z fee post multiplied because
of rotation with respect to current taxes.
Then we multiply that by four with our Twitter times.
Are the fee print multiplied because of rotation with respect to fix the taxes?
Then we post multiply our zip over r t r that fi with r y better post-meal blood because of rotation
with respect to current y axis.
Then we print multiply.
Our built up with our z over times are Twitter.
Times are Z.
Three times are y better pray multiplied because of rotation with respect to fixed x axis.
So whole rotation is R equals two are expelled at times.
Are that over times?
Our Twitter times are the three times are y better.
This is very important to understand the rotation mattresses because it is not only needed in robotics,
but also in mechanics, geometry, 3D design and so on.
See you on the next lesson.