Supplementary Material: Rates of Convergence

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來自 University of Pennsylvania 的課程

Robotics: Aerial Robotics

1665 個評分

University of Pennsylvania

Robotics: Aerial Robotics

1665 個評分

課程 1（共 6 門，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.

從本節課中

Introduction to Aerial Robotics

Welcome to Week 1! In this week, you will be introduced to the exciting field of Unmanned Aerial Robotics (UAVs) and quadrotors in particular. You will learn about their basic mechanics and control strategies and realize how careful component selection and design affect the vehicles' performance. This week also provides you with instructions on how to download and install Matlab. This software will be used throughout this course in exercises and assignments, so it is strongly recommended to familiarize yourself with Matlab soon. Tutorials to help you get started are also provided in this week.

Basic Mechanics4:27

Dynamics and 1-D Linear Control10:29

Design Considerations3:37

Design Considerations (continued)4:33

Agility and Maneuverability7:00

Component Selection4:24

Effects of Size4:38

Supplementary Material: Introduction3:19

Supplementary Material: Dynamical Systems3:15

Supplementary Material: Rates of Convergence3:20

與講師見面

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

In this segment, we'll examine the rate of convergence of the error function to 0.

Recall the control problem we looked at in the lecture.

We wanted to determine the appropriate input that will cause the error between

a desired state and the actual state of a dynamical system to eventually become 0.

Mathematically, we want the error function e equals x desired minus x

to go to 0 as t approaches infinity.

However, we specified a second criteria that was a bit stricter.

Not only do we want the error function to eventually reach 0,

we want it to reach 0 at a certain rate.

More formally we say that a function exponentially converges to 0 if there

exists constants alpha and beta, and a constant time t0,

such that for all t greater than t0, the norm of the error function

is always less than an exponential function defined by alpha and beta.

Note than many values of alpha, beta and

t0 may satisfy this criteria for a given error function.

We only need to find one such function.

On the other hand, not all values of alpha,

beta and t0 will satisfy this criteria.

Again, we only need to find one set of values that do fulfill the criteria.

Throughout this course we will use a PD or PID controller as our method for

controlling a quadrotor.

In lecture, we discuss the behavior of this controller under different values of

Kv and Kp in the context of controlling the height of a quadrotor.

For a well tune PD controller we saw our response curve

that looked like the one you see here.

We can calculate the error of such a curve by subtracting the actual height

at each point in time from the desired height.

The result is the curve illustrated here.

We can see that the curve clearly converges to 0,

indicating that our system eventually reaches its desired height.

For exponential convergence, we are concerned with

the magnitude of the error function and want to see how quickly it goes to 0.

On the plot on the right, the red curve plus the absolute value of the error over

time, we see the error is bounded by an exponential function, the dash curve.

This exponential curve has the constants t0 equals 0,

alpha equals three-halves, and beta equals 1.

After times 0, the magnitude of the error remains

less than the function three-halves e to the negative t for all time.

Consider now the response we saw for a controller with a high proportional gain.

In this case, the error function oscillates significantly

before approaching the desired value of 0.

Again, we want to plot the absolute value of the error function.

It turns out that even though the error oscillates,

we can still find an exponential curve that serves as an upper bound for

the absolute value of the error function.

First, this oscillatory error function also converges exponentially to 0.

These examples exemplify how for a PD controller,

even though the error function may have different characteristics,

it will always converge exponentially to 0 provided that Kp and Kv are positive.

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