Supplementary Material: Skew-Symmetric Matrices and the Hat Operator

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Robotics: Aerial Robotics

1846 calificaciones

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University of Pennsylvania

Robotics: Aerial Robotics

1846 calificaciones

Curso 1 de 6 en SpecializationRobotics

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.

De la lección

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.

Supplementary Material: Rigid-Body Displacements6:53

Supplementary Material: Properties of Functions3:08

Supplementary Material: Symbolic Calculations in Matlab4:31

Supplementary Material: The atan2 Function2:47

Supplementary Material: Eigenvalues and Eigenvectors of Matrices5:42

Supplementary Material: Quaternions2:55

Supplementary Material: Matrix Derivative1:33

Supplementary Material: Skew-Symmetric Matrices and the Hat Operator4:21

Conoce a los instructores

Vijay Kumar
Nemirovsky Family Dean of Penn Engineering and Professor of Mechanical Engineering and Applied Mechanics
School of Engineering and Applied Science

In this lecture, we'll talk about properties of skew-symmetric matrices and

the hat operator, which we saw in the expressions for angular velocity.

First let's define an operation called the matrix transpose,

denoted by the superscript T.

Let A be an n by m matrix, and let A i,

j denote the element in the ith row and jth column of A.

The i, jth component of the transpose of A is the j, ith component of A.

Loosely speaking, the transpose operation flips the rows columns of the matrix A.

Consider the following two by two matrix.

The transpose of this matrix swaps the terms in the 2, 1 and 1, 2 positions.

Note that the diagonal of the matrix always remains unchanged

after taking the transpose.

Consider the following rectangular matrix.

The transpose of this matrix looks like this.

Notice how for rectangular matrices, the dimensions of the transpose is different

from the dimensions of the original matrix.

In this example, the original matrix is a two by three matrix.

The transpose becomes a three by two matrix.

Consider the matrix that is the result of the previous problem.

The transpose of this matrix is calculated here.

Notice that we get back to the original matrix from the last problem.

We see from this example that the transpose of A transpose is A itself.

Finally, consider the one by three matrix which is simply a row vector.

The transpose operation turns this row vector into a column vector.

We say that a matrix A is symmetric is A transpose = A.

We say that a matrix A is skew-symmetric if A transpose = -A.

Let's consider specifically 3x3 skew-symmetric matrices.

Consider an arbitrary 3x3 matrix A.

Using the definition of skew-symmetric,

this matrix is skew-symmetric if the following expression is satisfied.

Matching up the components of the two matrices on either side of the expression,

we get six constraints that must be satisfied for a to be skew symmetric.

We see that the first three constraints say that the components A11, A22 and

A33 must equal the negative of themselves.

The only scalar that satisfies this criteria is 0.

We see that a skew-symmetric matrix must have all 0s on its diagonal.

The last three constraints relate the remaining six components of the matrix.

Substituting these constraints into the matrix gives us the following general

expression for a 3x3 skew-symmetric matrix.

In particular, notice that because of the constraints for skew symmetry,

this matrix only has three independent parameters.

As a result, we can concisely represent any skew symmetric

3x3 matrix as a 3x1 vector.

The hat operator allows us to switch between these two representations.

Exquisitely, A Hat or A is a three by one vector, it's a three

by three skew-symmetric matrix defined by the three components of the vector A.

For example, consider the vector, omega = 1, 2, 3.

The corresponding skew-symmetric matrix, omega hat is shown here.

The hat operator is also used to denote the cross product between two vectors.

That is u cross v can be written as u hat times v.

This means that the cross product of u and

v = to the skew symmetric matrix corresponding to u x v.

For this reason,

the skew symmetric matrix corresponding to u hat is sometimes denoted as u cross.

In lecture, we define the following two angular velocity vectors.

Will these angular velocity vectors always exist?

Recall that we proved that R transpose R dot, and

R dot R transpose are both skew symmetric matrices.

From the properties we derived earlier,

we then note that we are guaranteed to be able to find angular velocity vectors,

omega b and omega s, that satisfy the given definitions of angular velocity.

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