Supplementary Material: State-Space Form

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En provenance du cours de University of Pennsylvania

Robotics: Aerial Robotics

1881 notes

University of Pennsylvania

Robotics: Aerial Robotics

1881 notes

Cours 1 sur 6 dans Specialization Robotics

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.

À partir de la leçon

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.

Formulation3:18

Newton-Euler Equations6:05

Principal Axes and Principal Moments of Inertia6:03

Quadrotor Equations of Motion5:13

Supplementary Material: State-Space Form4:32

Supplementary Material: Getting Started With the First Programming Assignment3:22

Rencontrer les enseignants

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

In lecture, we saw the dynamical equations with a quadrotor written as matrix

equations, which we call state-space form.

In this segment, we'll demonstrate how to transform the ordinary differential

equations representing a dynamical system into state-space form.

Recall that a dynamical system is a system where the effects of

actions do not immediately affect the system.

We saw that the evolution of the states of these systems

is governed by a set of ordinary differential equations.

It's often helpful in control problems to rearrange

these ordinary differential equations into state-space form.

This means that we represent the differential equations in the form

x = f(x, u).

Where x is a matrix of states and u is a matrix of inputs.

We can do this in a very systematic manner.

Suppose we have an ordinary differential equation

governing a one dimensional system whose position is represented by y.

We first identify the order of the system n.

Recall that the order is the highest derivative that

appears in the differential equation.

We then define the states x1, which is the position y, and x2 the first derivative of

y, and so on until we define a state for the n minus first derivative of y.

Next, we create the state vector x,

which is a vector containing the previously defined states.

These states are governed by the following set of coupled first-order

differential equations.

We see that because of the way we've defined the states,

the first equation simply states that the derivative of x1 is x2.

The second equation tells us that the derivative of x2 is x3.

The only non-trivial differential equation is the derivative of xn

which could be a function of all the other states plus the input u.

We get this function by rearranging the governing ordinary differential equation.

Finally, we stock these first-order differential equations into a matrix.

On the left hand side, we have the matrix x dot.

On the right hand side we have a matrix whose components are functions of

the state's x and the input u.

Consider the Mass-Spring system we previously examined.

Governed by the shown ordinary differential equation.

The highest derivative that appears in this equation is the second derivative,

making this a second order system.

We need to define the states x1=y and x2=y dot.

Next, we create the state vector x, in this case x contains only two components,

y and y dot which we have designated as x1 and x2.

We can now define the system of first-order or differential equations.

The first equation is trivial and simply states that the derivative of x1 is x2.

We get the second equation from solving for

y double dot using the governance ordinary differential equation.

We see that the derivative of x2 is a function of x1 and u.

We can write these two differential equations as a matrix.

We see that because k and

m are constants, the system's actually linear in the states and input.

As a result, we can write the equations in the following manner.

We see that this equation is in the form x dot = Ax + Bu,

which is the general form for a linear state's base equation.

Again the matrix equation is linear and the state's x and the input's u.

We can demonstrate how to extend this procedure to higher order systems

by using the Planar Quadrotor Model.

From lecture we know the Planar Quadrotor is governed by the following set of

ordinary differential equations which are written in the terms of the variables y,

z and phi.

Here, the highest derivative appearing in any differential equation

is the second derivative.

So this is still a second order system.

Next, we essentially must carry out the required steps for each variable.

Because n equals two, we need define states for the position and velocity of y,

z, and phi.

This gives us a system with six states.

These six states are placed into one state vector.

We can now define the system of first-order differential equations.

Again, the first three equations simply relate states to each other.

The last three equations come from rearranging the three governing ordinary

differential equations.

We can now place these equations into a single matrix equation.

These equations are non linear because of the functions sin and

cosine of the state x three.

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