I'm Davide and I have a problem with the derivation of a function that later should be given to ode15i in Matlab.
Basically I've derived a big symbolic expression that describe the motion of a spececraft with a flexible appendice (like a solar panel). My goal is to obtain a function handle that can be integrated using the built-in implicit solver in Matlab (ode15i).
The problem I've encounter is the slowness of the Symbolic computations, especially in the function "daeFunction" (I've run it and lost any hope for a responce after 3/4 hours had passed).
The system of equations, that is derived using the Lagrange's method is an implicit ode.
The complex nature of the system arise from the flexibility modelling of the solar panel.
I am open to any suggestions that would help me in:
running the code properly.
running it as efficiently as possible.
Thx in advance.
Here after I copy the code. Note: Matlab r2021a was used.
close all
clear
clc
syms t
syms r(t) [3 1]
syms angle(t) [3 1]
syms delta(t)
syms beta(t) [3 1]
mu = 3.986e14;
mc = 1600;
mi = 10;
k = 10;
kt = 10;
Ii = [1 0 0 % for the first link it is different thus I should do a functoin or something that writes everything into an array or a vector
0 5 0
0 0 5];
% Dimension of satellite
a = 1;
b = 1.3;
c = 1;
Ic = 1/12*mc* [b^2+c^2 0 0
0 c^2+a^2 0
0 0 a^2+b^2];
ra_c = [0 1 0]';
a = diff(r,t,t);
ddelta = diff(delta,t);
dbeta = diff(beta,t);
dddelta = diff(delta,t,t);
ddbeta = diff(beta,t,t);
R= [cos(angle1).*cos(angle3)-cos(angle2).*sin(angle1).*sin(angle3) sin(angle1).*cos(angle3)+cos(angle2).*cos(angle1).*sin(angle3) sin(angle2).*sin(angle3)
-cos(angle1).*sin(angle3)-cos(angle2).*sin(angle1).*cos(angle3) -sin(angle1).*sin(angle3)+cos(angle2).*cos(angle1).*cos(angle3) sin(angle2).*cos(angle3)
sin(angle2).*sin(angle3) -sin(angle2).*cos(angle1) cos(angle2)];
d_angle1 = diff(angle1,t);
d_angle2 = diff(angle2,t);
d_angle3 = diff(angle3,t);
dd_angle1 = diff(angle1,t,t);
dd_angle2 = diff(angle2,t,t);
dd_angle3 = diff(angle3,t,t);
d_angle = [d_angle1;d_angle2;d_angle3];
dd_angle = [dd_angle1;dd_angle2;dd_angle3];
omega = [d_angle2.*cos(angle1)+d_angle3.*sin(angle2).*sin(angle1);d_angle2.*sin(angle1)-d_angle3.*sin(angle2).*cos(angle1);d_angle1+d_angle3.*cos(angle2)]; % this should describe correctly omega_oc
d_omega = diff(omega,t);
v1 = diff(r1,t);
v2 = diff(r2,t);
v3 = diff(r3,t);
v = [v1; v2; v3];
[J,r_cgi,R_ci]= Jacobian_Rob(4,delta,beta);
% Perform matrix multiplication
for mm = 1:4
vel(:,mm) = J(:,:,mm)*[ddelta;dbeta];
end
vel = formula(vel);
dr_Ccgi = vel(1:3,:);
omega_ci = vel(4:6,:);
assumeAlso(angle(t),'real');
assumeAlso(d_angle(t),'real');
assumeAlso(dd_angle(t),'real');
assumeAlso(r(t),'real');
assumeAlso(a(t),'real');
assumeAlso(v(t),'real');
assumeAlso(beta(t),'real');
assumeAlso(delta(t),'real');
assumeAlso(dbeta(t),'real');
assumeAlso(ddelta(t),'real');
assumeAlso(ddbeta(t),'real');
assumeAlso(dddelta(t),'real');
omega = formula(omega);
Tc = 1/2*v'*mc*v+1/2*omega'*R*Ic*R'*omega;
% kinetic energy of all appendices
for h = 1:4
Ti(h) = 1/2*v'*mi*v+mi*v'*skew(omega)*R*ra_c+mi*v'*skew(omega)*R*r_cgi(:,h)+mi*v'*R*dr_Ccgi(:,h)+1/2*mi*ra_c'*R'*skew(omega)'*skew(omega)*R*ra_c ...
+ mi*ra_c'*R'*skew(omega)'*skew(omega)*R*r_cgi(:,h)+mi*ra_c'*R'*skew(omega)'*R*dr_Ccgi(:,h)+1/2*omega'*R*R_ci(:,:,h)*Ii*(R*R_ci(:,:,h))'*omega ...
+ omega'*R*R_ci(:,:,h)*Ii*R_ci(:,:,h)'*omega_ci(:,h)+1/2*omega_ci(:,h)'*R_ci(:,:,h)*Ii*R_ci(:,:,h)'*omega_ci(:,h)+1/2*mi*r_cgi(:,h)'*R'*skew(omega)'*skew(omega)*R*r_cgi(:,h)+mi*r_cgi(:,h)'*R'*skew(omega)'*R*dr_Ccgi(:,h)...
+ 1/2*mi*dr_Ccgi(:,h)'*dr_Ccgi(:,h);
Ugi(h) = -mu*mi/norm(r,2)+mu*mi*r'/(norm(r,2)^3)*(R*ra_c+R*R_ci(:,:,h)*r_cgi(:,h));
end
Ugc = -mu*mc/norm(r,2);
Ue = 1/2*kt*(delta)^2+sum(1/2*k*(beta).^2);
U = Ugc+sum(Ugi)+Ue;
L = Tc + sum(Ti) - U;
D = 1/2 *100* (ddelta^2+sum(dbeta.^2));
%% Equation of motion derivation
eq = [diff(jacobian(L,v),t)'-jacobian(L,r)';
diff(jacobian(L,d_angle),t)'-jacobian(L,angle)';
diff(jacobian(L,ddelta),t)'-jacobian(L,delta)'+jacobian(D,ddelta)';
diff(jacobian(L,dbeta),t)'-jacobian(L,beta)'+jacobian(D,dbeta)'];
%% Reduction to first order sys
[sys,newVars,R1]=reduceDifferentialOrder(eq,[r(t); angle(t); delta(t); beta(t)]);
DAEs = sys;
DAEvars = newVars;
%% ode15i implicit solver
pDAEs = symvar(DAEs);
pDAEvars = symvar(DAEvars);
extraParams = setdiff(pDAEs,pDAEvars);
f = daeFunction(DAEs,DAEvars,'File','ProvaSum');
y0est = [6778e3 0 0 0.01 0.1 0.3 0 0.12 0 0 0 7400 0 0 0 0 0 0 0 0]';
yp0est = zeros(20,1);
opt = odeset('RelTol', 10.0^(-7),'AbsTol',10.0^(-7),'Stats', 'on');
[y0,yp0] = decic(f,0,y0est,[],yp0est,[],opt);
% Integration
[tSol,ySol] = ode15i(f,[0 0.5],y0,yp0,opt);
%% Funcitons
function [J,p_cgi,R_ci]=Jacobian_Rob(N,delta,beta)
% Function to compute Jacobian see Robotics by Siciliano
% N total number of links
% delta [1x1] beta [N-1x1] variable that describe position of the solar
% panel elements
beta = formula(beta);
L_link = [1 1 1 1]'; % Length of each link elements in [m], later to be derived from file or as function input
for I = 1 : N
A1 = Homog_Matrix(I,delta,beta);
A1 = formula(A1);
R_ci(:,:,I) = A1(1:3,1:3);
if I ~= 1
p_cgi(:,I) = A1(1:3,4) + A1(1:3,1:3)*[1 0 0]'*L_link(I)/2;
else
p_cgi(:,I) = A1(1:3,4) + A1(1:3,1:3)*[0 0 1]'*L_link(I)/2;
end
for j = 1:I
A_j = formula(Homog_Matrix(j,delta,beta));
z_j = A_j(1:3,3);
Jp(:,j) = skew(z_j)*(p_cgi(:,I)-A_j(1:3,4));
Jo(:,j) = z_j;
end
if N-I > 0
Jp(:,I+1:N) = zeros(3,N-I);
Jo(:,I+1:N) = zeros(3,N-I);
end
J(:,:,I)= [Jp;Jo];
end
J = formula(J);
p_cgi = formula(p_cgi);
R_ci = formula(R_ci);
end
function [A_CJ]=Homog_Matrix(J,delta,beta)
% This function is made sopecifically for the solar panel
% define basic rotation matrices
Rx = #(angle) [1 0 0
0 cos(angle) -sin(angle)
0 sin(angle) cos(angle)];
Ry = #(angle) [ cos(angle) 0 sin(angle)
0 1 0
-sin(angle) 0 cos(angle)];
Rz = #(angle) [cos(angle) -sin(angle) 0
sin(angle) cos(angle) 0
0 0 1];
if isa(beta,"sym")
beta = formula(beta);
end
L_link = [1 1 1 1]'; % Length of each link elements in [m], later to be derived from file or as function input
% Rotation matrix how C sees B
R_CB = Rz(-pi/2)*Ry(-pi/2); % Clarify notation: R_CB represent the rotation matrix that describe the frame B how it is seen by C
% it is the same if it was wrtitten R_B2C
% becouse bring a vector written in B to C
% frame --> p_C = R_CB p_B
% same convention used in siciliano how C sees B frame
A_AB = [R_CB zeros(3,1)
zeros(1,3) 1];
A_B1 = [Rz(delta) zeros(3,1)
zeros(1,3) 1];
A_12 = [Ry(-pi/2)*Rx(-pi/2)*Rz(beta(1)) L_link(1)*[0 0 1]'
zeros(1,3) 1];
if J == 1
A_CJ = A_AB*A_B1;
elseif J == 0
A_CJ = A_AB;
else
A_CJ = A_AB*A_B1*A_12;
end
for j = 3:J
A_Jm1J = [Rz(beta(j-1)) L_link(j-1)*[1 0 0]'
zeros(1,3) 1];
A_CJ = A_CJ*A_Jm1J;
end
end
function [S]=skew(r)
S = [ 0 -r(3) r(2); r(3) 0 -r(1); -r(2) r(1) 0];
end
I found your question beautiful. My suggestion is to manipulate the problem numerically. symbolic manipulation in Matlab is good but is much slower than numerical calculation. you can define easily the ode into a system of first-order odes and solve them using numerical integration functions like ode45. Your code is very lengthy and I couldn't manage to follow its details.
All the Best.
Yasien
I know this problem comes up a lot on here and I've read through many threads but am still stuck on my specific problem. I'm using a mass matrix with ode function to solve 4 equations simultaneously, 2 algebraic and 2 differential:
function ade
% Equation set
% y(1)== radius 1, y(2)== concentration Cc
% y(3)== radius 2 y(4)== concentration Cc2
%==========================================================================
clc
close all
M = [1 0 0 0
0 0 0 0
0 0 1 0
0 0 0 0];
options = odeset('Mass',M);
y0= [1,1,1,1];
tspan = [0:1:1440];
[t,y]= ode15s(#equations,tspan,y0,options);
function mainequations = equations (t, y)
m = 18;
m2 = 599;
ro = 1000;
ro2 = 1050;
k = 0.085;
k2 = 0.085;
theta = 46/8;
theta2 = 46/8;
De = 7.65*10^-14;
Cb = 0.993;
R = 8.29*10^-7;
alpha = (m*k)/ro;
alpha1 = (m2*k2)/ro2;
beta = (theta*k*y(2))/De;
beta2 = (theta2*k2*y(4))/De;
mainequations = [-alpha*y(2)
y(4) - y(2)-(beta * y(1)^2 *(1/y(1) - y(3)))
-alpha1*y(4)
Cb - y(4) -(beta * y(1)^2 + beta2*(y(3))^2)*(1/y(3) - 1/R)];
I'm pretty sure the problem is in the equations at the bottom, I'm guessing one is the wrong size, but when debugging all the y values are the same size i.e. 4x1. Can anyone help?
Thanks
I want to speed up my code. I always use vectorization. But in this code I have no idea how to avoid the for-loop. I would really appreciate a hint how to proceed.
thank u so much for your time.
close all
clear
clc
% generating sample data
x = linspace(10,130,33);
y = linspace(20,100,22);
[xx, yy] = ndgrid(x,y);
k = 2*pi/50;
s = [sin(k*xx+k*yy)];
% generating query points
xi = 10:5:130;
yi = 20:5:100;
[xxi, yyi] = ndgrid(xi,yi);
P = [xxi(:), yyi(:)];
% interpolation algorithm
dx = x(2) - x(1);
dy = y(2) - y(1);
x_ = [x(1)-dx x x(end)+dx x(end)+2*dx];
y_ = [y(1)-dy y y(end)+dy y(end)+2*dy];
s_ = [s(1) s(1,:) s(1,end) s(1,end)
s(:,1) s s(:,end) s(:,end)
s(end,1) s(end,:) s(end,end) s(end,end)
s(end,1) s(end,:) s(end,end) s(end,end)];
si = P(:,1)*0;
M = 1/6*[-1 3 -3 1
3 -6 3 0
-3 0 3 0
1 4 1 0];
tic
for nn = 1:numel(P(:,1))
u = mod(P(nn,1)- x_(1), dx)/dx;
jj = floor((P(nn,1) - x_(1))/dx) + 1;
v = mod(P(nn,2)- y_(1), dy)/dy;
ii = floor((P(nn,2) - y_(1))/dy) + 1;
D = [s_(jj-1,ii-1) s_(jj-1,ii) s_(jj-1,ii+1) s_(jj-1,ii+2)
s_(jj,ii-1) s_(jj,ii) s_(jj,ii+1) s_(jj,ii+2)
s_(jj+1,ii-1) s_(jj+1,ii) s_(jj+1,ii+1) s_(jj+1,ii+2)
s_(jj+2,ii-1) s_(jj+2,ii) s_(jj+2,ii+1) s_(jj+2,ii+2)];
U = [u.^3 u.^2 u 1];
V = [v.^3 v.^2 v 1];
si(nn) = U*M*D*M'*V';
end
toc
scatter3(P(:,1), P(:,2), si)
hold on
mesh(xx,yy,s)
This is the full example and is a cubic B-spline surface interpolation algorithm in 2D space.
I have this for loop
for mass = 1:.1:4
theta = 60;
Fw = [0 -9.8*mass 0];
Rw = [ 0.2*cosd(theta) 0.2*sind(theta) 0];
Mw = cross(Rw,Fw);
Fjrf = [30 50 0];
Rjrf = [0.4*cosd(theta) 0.4*sind(theta) 0];
Mjrf = cross(Rjrf,Fjrf);
alpha = 5;
Ialpha = [0 0 alpha*(0.02+mass*0.2^2)];
Ms = Ialpha - Mjrf - Mw;
omega = [0 0 pi];
alphav = [0 0 5];
anorm = ([-cosd(theta) -sind(theta) 0]*(norm(omega))^2);
atang = cross(Rjrf,alphav);
a = anorm + atang;
Fmuscle = norm(mass*a - Fjrf - Fw)
mass
plot(mass,Fmuscle)'
hold on
end
Where in the end the output is just various values of Fmuscle, that correspond to values of mass varrying from 1 to 4 at intervals of .1. I am trying to plot all these values on a graph (mass =x and Fmuscle = y) but I cannot figure out how to generate this.
With the code I have above I simply get an empty graph, even though I get all the right variables.
Thanks!
This what you want?
i = 0; %set initial index
clear x y; %clear these variables
for mass = 1:.1:4
i = i + 1; %increment index
theta = 60;
Fw = [0 -9.8*mass 0];
Rw = [ 0.2*cosd(theta) 0.2*sind(theta) 0];
Mw = cross(Rw,Fw);
Fjrf = [30 50 0];
Rjrf = [0.4*cosd(theta) 0.4*sind(theta) 0];
Mjrf = cross(Rjrf,Fjrf);
alpha = 5;
Ialpha = [0 0 alpha*(0.02+mass*0.2^2)];
Ms = Ialpha - Mjrf - Mw;
omega = [0 0 pi];
alphav = [0 0 5];
anorm = ([-cosd(theta) -sind(theta) 0]*(norm(omega))^2);
atang = cross(Rjrf,alphav);
a = anorm + atang;
Fmuscle = norm(mass*a - Fjrf - Fw);
x(i) = mass; % set the appropriate row of x vector
y(i) = Fmuscle; % same for y
end
plot(x, y);
Good day,
I have code that stretches and rotates a gaussian 2d pdf as such:
mu = [0 0];
Sigma = [1 0; 0 1];
Scale = [3 0; 0 1];
Theta = 10;
Rotate = [cosd(Theta) -sind(Theta); sind(Theta) cosd(Theta)];
Sigma = (Sigma*Scale)*Rotate
x1 = -100:1:100; x2 = -100:1:100;
[X1,X2] = meshgrid(x1,x2);
F = mvnpdf([X1(:) X2(:)],mu,Sigma);
F = reshape(F,length(x2),length(x1));
imshow(F*255)
Unfortunately, when I change Theta to a value other than 0, it says
SIGMA must be a square, symmetric, positive definite matrix. Can I know what's going on
If you consult the article on Wikipedia about the general elliptical version of the Gaussian 2D PDF, it doesn't look like you're rotating it properly. In general, the equation is:
Source: Wikipedia
where:
Usually, A = 1 and we'll adopt that here. The angle theta will rotate the PDF counter-clockwise, and so we can use this raw form of the equation over mvnpdf. Using your definitions and constants, it would become this:
x1 = -100:1:100; x2 = -100:1:100;
[X1,X2] = meshgrid(X1, X2);
sigma1 = 1;
sigma2 = 1;
scale1 = 3;
scale2 = 1;
sigma1 = scale1*sigma1;
sigma2 = scale2*sigma2;
Theta = 10;
a = ((cosd(Theta)^2) / (2*sigma1^2)) + ((sind(Theta)^2) / (2*sigma2^2));
b = -((sind(2*Theta)) / (4*sigma1^2)) + ((sind(2*Theta)) / (4*sigma2^2));
c = ((sind(Theta)^2) / (2*sigma1^2)) + ((cosd(Theta)^2) / (2*sigma2^2));
mu = [0 0];
A = 1;
F = A*exp(-(a*(X1 - mu(1)).^2 + 2*b*(X1 - mu(1)).*(X2 - mu(2)) + c*(X2 - mu(2)).^2));
imshow(F*255);
I see that you are trying to rotate the variances in 2D... You have to hit your sigma matrix with your transform on both sides of it since the "sigma" matrix is essentially a tensor.
mu = [0 0];
Sigma = [1 0; 0 1];
Scale = [10 0;0 1];
Theta = pi/3;
m = makehgtform('zrotate',Theta);
m = m(1:2,1:2);
Sigma = m*(Sigma*Scale)*m.';
x1 = -100:1:100;
x2 = -100:1:100;
[X1,X2] = meshgrid(x1,x2);
F = mvnpdf([X1(:) X2(:)],mu,Sigma);
F = reshape(F,length(x2),length(x1));
imshow(F*255)