Lab 0: MATLAB and ODE Solvers | ME 4173 Robot Kinematics

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Introduction

The following report will display the results and conclusions of an experiment to simulate the output of an inverse pendulum system in MATLAB. The objectives of this experiment were to review MATLAB programming and using MATLAB to simulate ODEs and systems.
Objectives

* Examine the basics of MATLAB

* Use MATLAB to simulate a system

* Use ODE solvers to numerically integrate the system over a set time period
Apparatus
The apparatus used in this experiment was MATLAB. It was used to provide a simulation environment to analyze the inverse pendulum’s motion.
Experiments and Results
There were six components of this experiment. This experiment was mostly familiarizing with MATLAB. All code used is illustrated in the Appendix – Code.

The first part consisted of learning commands within the MATLAB environment. It was a brief overview of how commands work in MATLAB. There was no code used in the part of the experiment.

The second part of the experiment examined how arrays were created and used in MATLAB. The first step was to create a matrix. This matrix was then subjected to various commands including eye( ), zeroes( ), and ones( ). Indexing was also used to access various parts of the matrix. Matrix operations such as transpose, inverse, size and length were also shown.

Part three of the experiment explained how a script was created and what it was used for. A script file was created and used for other parts of the lab.
Part four illustrated what a function was and how they are created. It explained how syntax was important in the naming and calling of the function. This allowed the user to write a function and save in within a directory. This function could be called by the function name and input variables and return an answer depending on what the function was coded to perform. An addTwo(a,b) function was created as an example.

Part five contained the method for utilizing MATLAB`s ordinary differential equation solver (ODE). It uses this solver to perform numerical integration which allows systems to be solved within the MATLAB environment. MATLAB contains a number of ODE solvers: ode23, ode113, ode15s, ode23s, ode23t, ode23tb, ode45. An example of an ODE solver was used to illustrate the process of utilizing the solver.

Part six of the experiment was the simulation of an inverted pendulum. The pendulum was simulated using the ODE solver. Before the ODE solver could be used, the ordinary differential equations for the inverted pendulum needed to be developed and placed into state space variables. This was done by creating a function called xdot( ). The function has two element: xdot(1) and xdot(2). The two were plotted using a plot( ) function in MATLAB. The result was two sinusoidal waves which were cyclic in nature.
Discussion
The first problem which arose concerned the trigonometric functions within MATLAB. The code was written in degrees but the answer was expected to be in radians. It was giving a response which looked like an analog signal which was growing exponentially. Once the code was changed to display in radians, there was another issue. The code was checked by the TA and verified to be correct but the response was not what was expected. The issue was actually within MATLAB. Another MATLAB application was used with the same code and the issue was resolved.
Conclusion
The experiment was much more difficult than anticipated but that was due to small errors in syntax and in the software itself. The expected results were achieved in all questions of the experiment. The analysis of the inverted pendulum was completed using the ODE solver within MATLAB. The plot revealed two undamped sinusoidal waves which represented the angular velocity and angle of the system with a specific input.

Appendix- MATLAB Code * Part 2
A = [1 2 3 4 5; 6 7 8 9 10];
A(2,2)

A(1,end)

x = [1 2 3 4 5] x =
1 2 3 4 5 x(1) x(end)

inv(A)
A’
size(A) length(x) * Part 4 function var = addTwo(a,b) var = a + b;

addTwo(1,2)

* Part 5
Function
function ydot = rates(t,y) ydot(1,1) = y(3); ydot(2,1) = y(4); ydot(3,1) = -0.1; ydot(4,1) = 0.2;

Plotting y0 = [0;0;0;0]; %initial conditions tInitial = 0; %initial time tFinal = 10; %final time
[tout, yout] = ode23(’rates’,[tInitial tFinal],y0); % call to the ode solver

* Part 6
Function
% Function xdot with u = 0

function xdot = func(t,x) m = 2;
M = 8;
L = 1; g = 9.8; xdot = zeros(2,1); xdot(1) = x(2); xdot(2) = 2*g*sind(x(1))-m*L/(2*(m+M))*x(2)^2*sind(2*x(1))/(4*L/3-m*L*cosd(x(1))^2/(M+m));

Plotting Results close all clear all clc x = [5,0];

[tOut , xOut] = ode45(@func, [0 60], x);

plot (tOut, xOut)