How is it possible that vpasolve gives empty vectors when there are solutions?

I have a system of equations containing cosine and sine functions. The unknowns are angles so the range is between 0 and 360 degrees. When I put in these ranges then vpasolve gives back empty vectors when there are solutions. How is it possible that vpasolve does not give back the solutions? And vpasolve gives back the solutions when I narrow the ranges down. But it should find the solutions within a range of at most 90 degrees.

This is my code:

clear; close all; clc

% Given parameters
A = [0 0]; % Coordinates of winch

SlingPos = [2   3;
            10  8;
            0  26;
            22 20;
            52 32;]; % Coordinates of the fixed point of the slings

SlingLen = [1;
            2;
            2.5;
            3;
            4]; % Length of the slings

Pwr = [0;
       0;
       0;
       0;
       0;]; % Percent friction in sheave 1

M = 500;  % Mass of load

% Calculate angles and resultant force in the slings
syms(sym('theta',[1 length(SlingLen)])) % Create symbolic variables for the angles theta
syms(sym('T',[1 length(SlingLen)])) % Create symbolic variables for the resultant forces T
syms(sym('phi',[1 length(SlingLen)])) % Create symbolic variables for the angles phi
Fg = M*9.81; % Gravity

Fpull = zeros(length(SlingLen),1);
Fpull(end) = ((100+Pwr(end))/100)*Fg; % Pulling force of the last sheave

% Calculate other pulling forces
for i = length(SlingLen)-1:-1:1
    Fpull(i) = ((100+Pwr(i))/100)*Fpull(i+1);
end

%TEST

[phi1, theta1, T1, phi2, theta2, T2, phi3, theta3, T3, phi4, theta4, T4, phi5, theta5, T5] = vpasolve(...
    -Fpull(1)*cosd(phi1)-T1*cosd(theta1)+Fpull(2)*cosd(phi2)==0,...
    Fpull(1)*sind(phi1)+T1*sind(theta1)-Fpull(2)*sind(phi2)==0,...
    -Fpull(2)*cosd(phi2)-T2*cosd(theta2)+Fpull(3)*cosd(phi3)==0,...
    Fpull(2)*sind(phi2)+T2*sind(theta2)-Fpull(3)*sind(phi3)==0,...
    -Fpull(3)*cosd(phi3)-T3*cosd(theta3)+Fpull(4)*cosd(phi4)==0,...
    Fpull(3)*sind(phi3)+T3*sind(theta3)-Fpull(4)*sind(phi4)==0,...
    -Fpull(4)*cosd(phi4)-T4*cosd(theta4)+Fpull(5)*cosd(phi5)==0,...
    Fpull(4)*sind(phi4)+T4*sind(theta4)-Fpull(5)*sind(phi5)==0,...
    -Fpull(5)*cosd(phi5)-T5*cosd(theta5)==0,...
    Fpull(5)*sind(phi5)+T5*sind(theta5)-Fg==0,...
    A(1,1)+sqrt((abs(A(1,1)-SlingPos(1,1))+SlingLen(1)*cosd(theta1))^2+(abs(A(1,2)-SlingPos(1,2))-SlingLen(1)*sind(theta1))^2)*cosd(phi1)==SlingPos(1,1)+SlingLen(1)*cosd(theta1),...
    SlingPos(1,1)+SlingLen(1)*cosd(theta1)+sqrt((abs(SlingPos(1,1)+SlingLen(1)*cosd(theta1)-SlingPos(2,1))+SlingLen(2)*cosd(theta2))^2+(abs(SlingPos(1,2)-SlingLen(1)*sind(theta1)-SlingPos(2,2))-SlingLen(2)*sind(theta2))^2)*cosd(phi2)==SlingPos(2,1)+SlingLen(2)*cosd(theta2),...
    SlingPos(2,1)+SlingLen(2)*cosd(theta2)+sqrt((abs(SlingPos(2,1)+SlingLen(2)*cosd(theta2)-SlingPos(3,1))-SlingLen(3)*cosd(theta3))^2+(abs(SlingPos(2,2)-SlingLen(2)*sind(theta2)-SlingPos(3,2))-SlingLen(3)*sind(theta3))^2)*cosd(phi3)==SlingPos(3,1)+SlingLen(3)*cosd(theta3),...
    SlingPos(3,1)+SlingLen(3)*cosd(theta3)+sqrt((abs(SlingPos(3,1)+SlingLen(3)*cosd(theta3)-SlingPos(4,1))+SlingLen(4)*cosd(theta4))^2+(abs(SlingPos(3,2)-SlingLen(3)*sind(theta3)-SlingPos(4,2))+SlingLen(4)*sind(theta4))^2)*cosd(phi4)==SlingPos(4,1)+SlingLen(4)*cosd(theta4),...
    SlingPos(4,1)+SlingLen(4)*cosd(theta4)+sqrt((abs(SlingPos(4,1)+SlingLen(4)*cosd(theta4)-SlingPos(5,1))+SlingLen(5)*cosd(theta5))^2+(abs(SlingPos(4,2)-SlingLen(4)*sind(theta4)-SlingPos(5,2))-SlingLen(5)*sind(theta5))^2)*cosd(phi5)==SlingPos(5,1)+SlingLen(5)*cosd(theta5),...
    [phi1,theta1,T1,phi2,theta2,T2,phi3,theta3,T3,phi4,theta4,T4,phi5,theta5,T5],...
    [0 360;0 360;-inf inf;0 360;0 360;-inf inf;0 360;0 360;-inf inf;0 360;0 360;-inf inf;0 360;0 360;-inf inf]);


PosSheave1 = [double(SlingPos(1,1)+SlingLen(1)*cosd(theta1)); double(SlingPos(1,2)-SlingLen(1)*sind(theta1))]
PosSheave2 = [double(SlingPos(2,1)+SlingLen(2)*cosd(theta2)); double(SlingPos(2,2)-SlingLen(2)*sind(theta2))]
PosSheave3 = [double(SlingPos(3,1)+SlingLen(3)*cosd(theta3)); double(SlingPos(3,2)-SlingLen(3)*sind(theta3))]
PosSheave4 = [double(SlingPos(4,1)+SlingLen(4)*cosd(theta4)); double(SlingPos(4,2)-SlingLen(4)*sind(theta4))]
PosSheave5 = [double(SlingPos(5,1)+SlingLen(5)*cosd(theta5)); double(SlingPos(5,2)-SlingLen(5)*sind(theta5))]

[Jeroen Staps]

My code does give a solution when I change the ranges to:

[270 360;0 90;-inf inf;270 360;90 270;-inf inf;180 270;0 90;-inf inf;270 360;0 90;-inf inf;20 28;143 150;-inf inf]

[Jeroen Staps]

But is does not give me a solution when the ranges are:

[270 360;0 90;-inf inf;270 360;90 270;-inf inf;180 270;0 90;-inf inf;270 360;0 90;-inf inf;0 90;90 180;-inf inf]

[Walter Roberson]

I notice that the right hand sides use theta1, theta2, theta3, theta4, theta5, but the left hand sides use theta1, theta1, theta2, theta3, theta4 -- with theta1 used twice and theta5 not used. Is that correct?

[Walter Roberson]

I see that your T* variables are all in the range -inf to +inf . Is it possible to narrow down the sign of those? The different signs lead to different solutions, so 3^5 possibilities each of which generates two solutions per variable, so you end up needing to follow over 400 possibilities just based upon the signs of the T* variables.

[Jeroen Staps]

I do not see what you mean with theta1 used twice and not theta5.

I can narrow it down to 0-2*Fg for the T*

[Walter Roberson]

    A(1,1)+sqrt((abs(A(1,1)-SlingPos(1,1))+SlingLen(1)*cosd(theta1))^2+(abs(A(1,2)-SlingPos(1,2))-SlingLen(1)*sind(theta1))^2)*cosd(phi1)==SlingPos(1,1)+SlingLen(1)*cosd(theta1),...
                                                            ^1^                                                    ^1^                                                   ^1^        
    SlingPos(1,1)+SlingLen(1)*cosd(theta1)+sqrt((abs(SlingPos(1,1)+SlingLen(1)*cosd(theta1)-SlingPos(2,1))+SlingLen(2)*cosd(theta2))^2+(abs(SlingPos(1,2)-SlingLen(1)*sind(theta1)-SlingPos(2,2))-SlingLen(2)*sind(theta2))^2)*cosd(phi2)==SlingPos(2,1)+SlingLen(2)*cosd(theta2),...
                                   ^1^                                              ^1^                                     ^2^                                           ^1^                                     ^2^                                                    ^2^ 
                               

After the == the first equation uses theta1, the second uses theta2, and so on through theta5.

Before the == the first equation uses theta1, then the second uses theta1, third uses theta2, and so on. The last equation uses theta4, with there being no corresponding equation using theta5.

Perhaps this is to be expected for your situation; it looked a bit asymmetric to me.

[Jeroen Staps]

No that is not the problem, that is to be expected. Because A is the starting point and does not correspond to a angle. From there on SlingPos(1,1) has theta1 SlingPos(2,1) has theta 2 and so on.

Answer 1

[Walter Roberson]

vpasolve() does a modified newton search.

• This can involve projecting outside of any given bounds even though the solution is safely within bounds. It is a property of newton type searches that in some cases as the search approaches the correct location it can end up projecting quite a distance away. When vpasolve() passes a boundary, it immediately gives up, and does not attempt to bounce off of the boundary
• vpasolve() uses the array bounds as the initial points if they are finite. In situations where there might be multiple zero crossings, which is common with trig, then this can result in the initial points having the same sign, causing it to think there is no need to look for a zero crossing
• because the search is more or less following local gradient, it is possible to get caught in a local minima. In such cases, vpasolve() will give up.