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Cart_Pendulum.m
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Cart_Pendulum.m
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clc;
close all;
clear all;
1;
pkg load control
pkg load symbolic
##**************************************************************************
##* OCTAVE PROGRAMMING (e-Yantra 2019-20)
##* ====================================
##* This software is intended to teach Octave Programming and Mathematical
##* Modeling concepts
##* Theme: Biped Patrol
##* Filename: Cart_Pendulum.m
##* Version: 1.0.0
##* Date: November 3, 2019
##*
##* Team ID :5179
##* Team Leader Name:Sahil Garg
##* Team Member Name:Sahil Garg, Manan Anand, Drishti Agarwal, Priyanshi Gupta
##*
##*
##* Author: e-Yantra Project, Department of Computer Science
##* and Engineering, Indian Institute of Technology Bombay.
##*
##* Software released under Creative Commons CC BY-NC-SA
##*
##* For legal information refer to:
##* http://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
##*
##*
##* This software is made available on an “AS IS WHERE IS BASIS”.
##* Licensee/end user indemnifies and will keep e-Yantra indemnified from
##* any and all claim(s) that emanate from the use of the Software or
##* breach of the terms of this agreement.
##*
##* e-Yantra - An MHRD project under National Mission on Education using
##* ICT(NMEICT)
##*
##**************************************************************************
## Function : draw_cart_pendulum()
## ----------------------------------------------------
## Input: y - State Vector. In case of inverted cart pendulum, the state variables
## are position of cart x, velocity of cart x_dot, angle of pendulum
## bob theta wrt vertical and angular velocity theta_dot of pendulum
## bob.
##
## Purpose: Takes the state vector as input. It draws the inverted cart pendulum in
## a 2D plot.
function draw_cart_pendulum(y,m, M, L)
x = y(1);
theta = y(3);
W = 1*sqrt(M/5); # cart width
H = 0.5*sqrt(M/5); # cart height
wr = 0.2; # wheel radius
mr = 0.3*sqrt(m); # mass radius
y = wr/2 + H/2;
w1x = x - 0.9*W/2;
w1y = 0;
w2x = x + 0.9*W/2 - wr;
w2y = 0;
px = x + L*sin(theta);
py = y - L*cos(theta);
hold on;
clf;
axis equal;
rectangle('Position',[x-W/2,y-H/2,W,H],'Curvature',.1,'FaceColor',[1 0.1 0.1])
rectangle('Position',[w1x,w1y,wr,wr],'Curvature',1,'FaceColor',[0 0 0])
rectangle('Position',[w2x,w2y,wr,wr],'Curvature',1,'FaceColor',[0 0 0])
line ([-10 10], [0 0], "linestyle", "-", "color", "k");
line ([x px], [y py], "linestyle", "-", "color", "k");
rectangle('Position',[px-mr/2,py-mr/2,mr,mr],'Curvature',1,'FaceColor',[.1 0.1 1])
xlim([-6 6]);
ylim([-2 3]);
set(gcf, 'Position', [200 300 1000 400]);
drawnow
hold off
endfunction
## Function : cart_pendulum_dynamics()
## ----------------------------------------------------
## Input: y - State Vector. In case of inverted cart pendulum, the state variables
## are position of cart x, velocity of cart x_dot, angle of pendulum
## bob theta wrt vertical and angular velocity theta_dot of pendulum
## bob.
## m - Mass of pendulum bob
## M - Mass of cart
## L - Length of cart
## g - Acceleration due to gravity
## u - Input to the system. Input is the horizontal force acting on the cart.
##
## Output: dy - Derivative of State Vector.
##
## Purpose: Calculates the value of the vector dy according to the equations which
## govern this system.
function dy = cart_pendulum_dynamics(y, m, M, L, g, u)
theta = y(3);
theta_dot = y(4);
dy(1,1) = y(2);
dy(2,1) = (((m*g/2)*sin(2*theta)) - (m*(-L)*(theta_dot^2)*sin(theta)) + u)/(M+(m*(sin(theta))^2));
dy(3,1) = y(4);
dy(4,1) = (((M+m)*g*sin(theta)) + (u*cos(theta)) - (m/2*(-L)*sin(2*theta)*(theta_dot^2)))/((-L)*(M+(m*(sin(theta))^2)));
endfunction
## Function : sim_cart_pendulum()
## ----------------------------------------------------
## Input: m - Mass of pendulum bob
## M - Mass of cart
## L - Length of cart
## g - Acceleration due to gravity
## y0 - Initial Condition of system
##
## Output: t - Timestep
## y - Solution array
##
## Purpose: This function demonstrates the behavior of cart pendulum system without
## any external input (u).
## This integrates the system of differential equation from t0 = 0 to
## tf = 10 with initial condition y0
function [t,y] = sim_cart_pendulum(m, M, L, g, y0)
tspan = 0:0.1:10; ## Initialise time step
u = 0; ## No Input
[t,y] = ode45(@(t,y)cart_pendulum_dynamics(y, m, M, L, g, u),tspan,y0); ## Solving the differential equation
endfunction
## Function : cart_pendulum_AB_matrix()
## ----------------------------------------------------
## Input: m - Mass of pendulum bob
## M - Mass of cart
## L - Length of cart
## g - Acceleration due to gravity
##
## Output: A - A matrix of system
## B - B matrix of system
##
## Purpose: Declare the A and B matrices in this function.
function [A, B] = cart_pendulum_AB_matrix(m , M, L, g)
A = [0,1,0,0;0,0,49/25,0;0,0,0,1;0,0,147/25,0];
B = [0;1/5;0;1/10];
endfunction
## Function : pole_place_cart_pendulum()
## ----------------------------------------------------
## Input: m - Mass of pendulum bob
## M - Mass of cart
## L - Length of cart
## g - Acceleration due to gravity
## y_setpoint - Reference Point
## y0 - Initial Condition
##
## Output: t - Timestep
## y - Solution array
##
## Purpose: This function demonstrates the behavior of inverted cart pendulum with
## external input using the pole_placement controller
## This integrates the system of differential equation from t0 = 0 to
## tf = 10 with initial condition y0 and input u = -Kx where K is
## calculated using Pole Placement Technique.
function [t,y] = pole_place_cart_pendulum(m, M, L, g, y_setpoint, y0)
[A,B] = cart_pendulum_AB_matrix(m,M,L,g); ## Initialize A and B matrix
eigs = [-4,-3,-1.5,-1.3]; ## Initialise desired eigenvalues
K=place(A,B,eigs);
tspan = 0:0.1:10;
[t,y] = ode45(@(t,y)cart_pendulum_dynamics(y, m, M, L, g, -K*(y-y_setpoint)),tspan,y0);
endfunction
## Function : lqr_cart_pendulum()
## ----------------------------------------------------
## Input: m - Mass of pendulum bob
## M - Mass of cart
## L - Length of cart
## g - Acceleration due to gravity
## y_setpoint - Reference Point
## y0 - Initial Condition
##
## Output: t - Timestep
## y - Solution array
##
## Purpose: This function demonstrates the behavior of inverted cart pendulum with
## external input using the LQR controller
## This integrates the system of differential equation from t0 = 0 to
## tf = 10 with initial condition y0 and input u = -Kx where K is
## calculated using LQR Controller.
function [t,y] = lqr_cart_pendulum(m, M, L, g, y_setpoint, y0)
[A,B] =cart_pendulum_AB_matrix(m,M,g,L) ; ## Initialize A and B matrix
Q = eye(4); ## Initialise Q matrix
R = 0.00001; ## Initialise R
K =lqr(A,B,Q,R) ; ## Calculate K matrix from A,B,Q,R matrices
tspan = 0:0.1:10;
[t,y] =ode45(@(t,y)cart_pendulum_dynamics(y, m, M, L, g, -K*(y-y_setpoint)),tspan,y0);
endfunction
## Function : cart_pendulum_main()
## ----------------------------------------------------
## Purpose: Used for testing out the various controllers by calling their
## respective functions and observing the behavior of the system. Constant
## parameters like mass of cart, mass of pendulum bob etc are declared here.
function cart_pendulum_main()
m = 1;
M = 5;
L = 2;
g = 9.8;
y0 = [0; 0; pi + 0.8; 0];
y_setpoint = [0; 0; pi; 0];
##[t,y] = sim_cart_pendulum(m, M, L, g, y0);
[t,y] = pole_place_cart_pendulum(m, M, L, g, y_setpoint, y0);
##[t,y] = lqr_cart_pendulum(m, M, L, g, y_setpoint, y0);
for k = 1:length(t)
draw_cart_pendulum(y(k, :), m, M, L);
endfor
endfunction