(Review) Lagrangian Dynamics

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Из курса от партнера University of Pennsylvania

Robotics: Capstone

58 оценки

University of Pennsylvania

Robotics: Capstone

58 оценки

Курс 6 из 6 — Specialization Robotics

In our 6 week Robotics Capstone, we will give you a chance to implement a solution for a real world problem based on the content you learnt from the courses in your robotics specialization. It will also give you a chance to use mathematical and programming methods that researchers use in robotics labs. You will choose from two tracks - In the simulation track, you will use Matlab to simulate a mobile inverted pendulum or MIP. The material required for this capstone track is based on courses in mobility, aerial robotics, and estimation. In the hardware track you will need to purchase and assemble a rover kit, a raspberry pi, a pi camera, and IMU to allow your rover to navigate autonomously through your own environment Hands-on programming experience will demonstrate that you have acquired the foundations of robot movement, planning, and perception, and that you are able to translate them to a variety of practical applications in real world problems. Completion of the capstone will better prepare you to enter the field of Robotics as well as an expansive and growing number of other career paths where robots are changing the landscape of nearly every industry. Please refer to the syllabus below for a week by week breakdown of each track. Week 1 Introduction MIP Track: Using MATLAB for Dynamic Simulations AR Track: Dijkstra's and Purchasing the Kit Quiz: A1.2 Integrating an ODE with MATLAB Programming Assignment: B1.3 Dijkstra's Algorithm in Python Week 2 MIP Track: PD Control for Second-Order Systems AR Track: Assembling the Rover Quiz: A2.2 PD Tracking Quiz: B2.10 Demonstrating your Completed Rover Week 3 MIP Track: Using an EKF to get scalar orientation from an IMU AR Track: Calibration Quiz: A3.2 EKF for Scalar Attitude Estimation Quiz: B3.8 Calibration Week 4 MIP Track: Modeling a Mobile Inverted Pendulum (MIP) AR Track: Designing a Controller for the Rover Quiz: A4.2 Dynamical simulation of a MIP Peer Graded Assignment: B4.2 Programming a Tag Following Algorithm Week 5 MIP Track: Local linearization of a MIP and linearized control AR Track: An Extended Kalman Filter for State Estimation Quiz: A5.2 Balancing Control of a MIP Peer Graded Assignment: B5.2 An Extended Kalman Filter for State Estimation Week 6 MIP Track: Feedback motion planning for the MIP AR Track: Integration Quiz: A6.2 Noise-Robust Control and Planning for the MIP Peer Graded Assignment: B6.2 Completing your Autonomous Rover

Из урока

Week 4

In the MIP track, you will learn how to build a model of the mobile inverted pendulum using a Lagrangian formulation to get equations of motion. This will help you build a simulation of a physical MIP that you can test your control ideas on. In the AR track, you will learn to design a controller that allows the rover to move to any target position when given its pose. You will then use this controller to get the rover to follow an AprilTag that you hold.

(Review) Lagrangian Dynamics8:24

A4.1 Modeling a Mobile Inverted Pendulum (MIP)2:58

Познакомьтесь с преподавателями

Kostas Daniilidis
Professor of Computer and Information Science
School of Engineering and Applied Science
Sid Deliwala
Director, Electrical and Systems Engineering Labs and Lecturer, Electrical and Systems Engineering
Department of Electrical and Systems Engineering

Today we are going to go over a very useful tool

used by engineers to explicitly write out the dynamics as some simple systems.

And then go over a popular method of controlling those systems.

Just a word of warning, the math in this lecture is a little more involved than

many of the previous lectures.

But our intention in doing this is to give you practical engineering tools

that you can use to apply to real problems.

You likely learned about Newton's laws in your introductory physics course.

It tell us that mechanical systems of a second order differential equations.

Meaning differential equations,

where there are no more than two derivatives in it of any variable.

Newton's laws let us model the accelerations of robots.

It's functions of their speed, velocity, and external forces.

In a kinematic section of this course, you'll learn how to generate the equations

and motion for a system using free-body diagrams.

Free-body diagrams work well in simple systems but

often times robotic models use complicated geometries and coordinate systems.

It can make free-body diagrams cumbersome.

In these cases, writing out the components of the internal forces can be tedious.

Especially when using rotate coordinate systems which is pretty common when

modeling robot joints.

Often times, writing out the energy of such systems

is much simpler than writing out all the forces.

It turns out that there is another way of generating the equations of motions for

a system from the mechanical energy alone so

you don't need to write out all the forces.

This formulation is known as Lagrangian mechanics.

And is another equivalent way of expressing Newton's laws.

For systems that aren't acted upon by outside forces,

Lagrangian mechanics says that the equations of motion are given by taking

the kinetic energy subtracting potential energy and then applying a special

expression called the Euler-Lagrange operator and setting the result to zero.

One big advantage of Lagrangian mechanics is that since this process is so simple,

it is easy to automate in software, once you can write down the system's energy.

Let's go over this in more detail before we do a simple example.

The mechanical energy of a system is the sum of its kinetic energy,

the energy from its motion and potential energy, for

example the energy due to gravity or springs.

Even in complicated systems, the kinetic and

potential energies are often much easier to write out than the forces.

The difference between the kinetic and the potential energy has a name and

it's called the Lagrangian.

By applying the Euler-Lagrange operator to the Lagrangian and

setting the result to zero you get the equation summation.

The Euler-Lagrange operator simply takes the partial derivative of velocities,

differentiates that with respect to time and

subtracts away the partial derivative of positions.

The form of the Euler-Lagrange operator, and the reason all of this works, is

a consequence of the principle in physics called the principle of least action.

We don't have time to go into the details of it today, but

feel free to look it up online if you're interested.

Finally, all of this accounts for internal forces,

but if external forces are acting on the system all you need to do is replace

these forces with a zero at the end.

As a quick example of how to use Lagrangian mechanics

let's derive the equations in motion for a simple pendulum.

A simple pendulum consists of a mass m attached to a rod of length l

that is free to rotate around sub-stationary pivot point.

Gravity pulls mass downwards.

Let's say that when a rod is vertically downwards it has an angle of

zero radiance.

So the state of the system is given by the rod angle and a rod angular velocity.

The first step is to rate out the kinetic and potential energy for the system.

The kinetic energy is given by one half a moment of inertia

times the angular velocity squared.

Which simplifies to one-half the mass times the rod length squared

times the angle of the velocity squared.

The potential energy is simply the energy due to gravity which is equal

to the negative of the mass times the rod length

times the acceleration due to gravity times the cosine of the rod angle.

Lagrangian is then the difference between the kinetic and the potential energy.

Let's apply the Euler-Lagrange operator to the system turn by turn.

The partial derivative of the Lagrangian with respect to position is equal to mgl

sine theta.

Now in this case we get a scalar but if the system had multiple degrees of

freedom, we would instead get a vector whenever we take a partial derivative.

The partial derivative of the Lagrangian with respect to velocity is equal to ml

squared times the angular velocity, and differentiating with respect to time,

gives ml squared times the angular acceleration.

We get the equations of motion when we combine these two terms and

set the result to zero.

Notably when we simplify the equations we see that the mass cancels out

leading to the interesting result that the frequency that the simple pendulum is

unaffected by its mass.

The equations in motion for a system resulting from the Euler-Lagrange operator

of a nice structure that can be useful to understand.

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