How can we create agile micro aerial vehicles that are able to operate autonomously in cluttered indoor and outdoor environments? You will gain an introduction to the mechanics of flight and the design of quadrotor flying robots and will be able to develop dynamic models, derive controllers, and synthesize planners for operating in three dimensional environments. You will be exposed to the challenges of using noisy sensors for localization and maneuvering in complex, three-dimensional environments. Finally, you will gain insights through seeing real world examples of the possible applications and challenges for the rapidly-growing drone industry. Mathematical prerequisites: Students taking this course are expected to have some familiarity with linear algebra, single variable calculus, and differential equations. Programming prerequisites: Some experience programming with MATLAB or Octave is recommended (we will use MATLAB in this course.) MATLAB will require the use of a 64-bit computer.
Euler Angles
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Robotics: Aerial Robotics
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Geometry and Mechanics
Welcome to Week 2 of the Robotics: Aerial Robotics course! We hope you are having a good time and learning a lot already! In this week, we will first focus on the kinematics of quadrotors. Then, you will learn how to derive the dynamic equations of motion for quadrotors. To build a better understanding on these notions, some essential mathematical tools are discussed in supplementary material lectures. In this week, you will also complete your first programming assignment on 1-D quadrotor control. If you have not done so already, please download, install, and learn about Matlab before starting the assignment.
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Vijay KumarNemirovsky Family Dean of Penn Engineering and Professor of Mechanical Engineering and Applied Mechanics
School of Engineering and Applied Science
Euler showed that three coordinates are necessary to describe a general rotation.
And these coordinates are called the Euler angles.
To show how Euler angles work, I want you to think about three successive rotations,
the first rotation going from frame A to frame B,
the second rotation going from B to C, and the third going from C to D.
The key idea behind Euler angles is that if I want to describe the rotation
A to D, I can break it up into three successive rotations,
A to B, B to C, and then C to D.
And you do this by simply multiplying the intermediate rotations.
You take the rotation from B to A, and then C to B, and then D to C,
and you end up on the left hand side with the rotation that goes from D to A.
So in this picture, you see that the first rotation about a1
is through the angle psi.
It's a rotation about the x-axis.
And that allows you to get to b1, b2, b3.
The second rotation is about the vector b2.
And that gets you to c1, c2, and c3.
That second rotation is through the angle phi.
And then finally, the third rotation,
which is about the vector c3, and that's through the angle theta.
So by simply multiplying rotations about the x-axis through psi,
the y-axis through phi, and the z-axis through theta,
you end up getting the net rotation, which is from frame a to frame d.
Again, you want to remember that the nomenclature here
refers both to displacements and to transformations.
So if I look at a rotation with a superscript A and
a subscript D, it serves two purposes.
It can transform vectors written in terms of D1, D2,
D3 into a vector whose components are written in terms of A1, A2, A3.
It can also refer to the fact that a rigid body has been displaced
from its initial orientation at frame A into its new orientation at frame D.
The three angles that you see here are roll, pitch, and yaw angles.
Imagine a vehicle whose axis is oriented along a1,
the first rotation is a roll rotation.
The second rotation is a pitching motion about the second axis, which is b2.
The third angle is a yaw rotation, and
that is about the third axis, which is c3.
Euler said that any rotation can be described by three successive rotations
about linearly independent axes.
So if you have three Euler angles, like the angles I just described to you,
you can deduce from them a 3 x 3 rotation matrix.
The question you want to ask yourself is if the reverse is true.
In other words, for every rotation matrix, is there a unique set of Euler angles?
In other words, does this map between the three coordinates,
the three Euler angles, and the rotation matrix, is this map one to one?
And the answer is no.
It's almost one to one, but
there are certain points that are analogous to the North pole and the South
pole on the Earth's surface, at which the Euler angles are not well defined.
This set of Euler angles is often called the X-Y-Z Euler angles.
And that's because the sequence in which these rotation
matrices are multiplied relates to a rotation about the x-axis,
followed by the y-axis, followed by the z-axis.
You can also have other types of Euler angles,
this particular one is called a Z-Y-Z Euler angles.
So again, the first rotation is about the z-axis.
You rotate about the z-axis through phi.
And then the second rotation is about the y-axis through theta.
And then the third rotation is a rotation about the z-axis through psi.
Notice that every rotation occurs about a body-fixed axis.
Once you complete the first rotation,
the second rotation about the y-axis is now about a new y-axis.
Because the first rotation rotated the original y-axis into a new position.
Although you have two rotations about z-axes,
the question you have to ask is, are they the same z-axis or different?
In other words, are the three axes linearly independent?
And you can see that the two z-axes are not co-linear, they are independent.
If this condition is satisfied, then the three angles
are Euler angles, and they can parameterize the set of rotations.
Note that if you have an Euler angle that is 0,
in this case, theta being equal to 0, you might run into problems.
So, theta equal to 0 would cause the two z-axes to be co-linear.
As a result, you will not have the condition of linear independence.
In other words, the three axes about which
you're performing rotations are no longer independent.
So theta equal to 0 is analogous to being at the North Pole or
the South Pole on the Earth's surface.
To explore this further, let's look inside the computations
that are involved in going from a rotation matrix to the three Euler angles.
I'm assuming I have a known rotation matrix, a set of nine numbers.
I want to recover the three Euler angles, phi, theta, and psi.
In order to do this, I can write out the result of the multiplication
of the three rotation matrices, the rotation about the z-axis,
followed by the rotation about the y-axis, followed by the rotation about the z-axis.
The first one through phi, the second one through theta, and
the third one through psi.
And then I can equate these two.
You will see that by equating the 33 element in the bottom right hand corner,
if I know R33, I can calculate what theta is.
So, that equation tells me what theta is.
Likewise, if I know what theta is, I can look at that 31 element, and
calculate what psi is.
In fact, there's a second equation that also gives me the same information.
The 32 element also tells me what psi is if I know what theta is.
And then finally, the 13 element and
the 23 elements tells me what phi is.
Knowing what theta is, I can calculate phi from these equations.
If the 33 element is non-zero, then we can go ahead and calculate theta.
You calculate it by taking the inverse cosine function.
Of course, the inverse cosine function has some ambiguity, and
because of that, you don't know the sine of theta.
It could either be positive or negative.
But modulo that ambiguity, you can determine theta.
And once you know theta, you can calculate psi and
phi using the inverse tangent function.
You will notice that we had two pieces of information for both psi and
for phi, and because of that, we don't use the standard inverse tangent function.
We use something called the a tan 2 function.
The a tan 2 function helps us overcome the ambiguity that exists
in inverse-tangent functions.
Unlike the traditional inverse-tangent function,
which only uses one equation to solve for an angle, here,
we use the fact that we have two equations for the same angle.
The inverse tangent function a tan 2 allows us to do this.
You will see from these equations that we have two sets of Euler angles.
And this is true for almost all rotation matrices.
These equations were derived,
assuming that the magnitude of R33 is not equal to 1.
It's less than 1.
So what happens if the magnitude of R33 is equal to 1?
It can either be plus 1 or -1.
In those cases, you can easily see that the angle theta
is going to be equal to either 0 or pi.
And you have two alternatives.
In both these cases, you will see from the grouping of terms that
the rotation matrices are functions only of the sum of two angles, phi and psi.
In other words, it's impossible to determine either phi or psi uniquely.
Given one, you can determine the other.
But it's impossible to disambiguate between phi and
psi because the groupings only contain terms that combine them.
So when R33 is equal to either plus 1 or
-1, you have infinite set of Euler angles.
In most of our work, we use a different set of Euler angles.
These are the Z-X-Y Euler angles.
And again, they're so called because the three rotation matrices we consider
are the rotation about the z-axis through psi,
followed by the rotation about the x-axis through phi, and
then the rotation of the y-axis through theta.
You should be able to verify by multiplying the rotations,
you get a rotation matrix that has the form shown in the slide.
We've seen for every set of Euler angles, you might have
at least two solutions for Euler angles for a given rotation matrix.
And at some points, you can have infinite solutions.
What you really want is a second set of Euler angles
to take care of the points at which you have infinite solutions.
So this suggests that you might have many,
many sets of Euler angles that you might want to consider so
that you don't have a point at which you have infinite solutions.
So the question that's worth asking is, what is the minimum number of sets of
Euler angles you need to cover all of the rotation group?
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