Solving trigonometric equations in MATLAB

Hi there,

I am trying to solve these two equations;

r1.cos(x1)=r2+r3.cos(x2)+r4.cos(x3)

r1.sin(x1)=r5+r3.sin(x2)+r4.sin(x3)

r1,r2,r3,r4,r5 and x1 are known, x2 and x3 are the unknowns. I need to obtain x2 and x3 values according to changing x1 value (between 100 and 126 degree).

I am a begginner for MATLAB, could you please help me out?

Thanks in advance.

Answer 1

[Birdman]

Try the following code. It should help you:

r1=1;r2=2;r3=3;r4=4;r5=5;%random values
x1=100:1:126;
syms x2 x3
for i=1:numel(x1)
    eq1=r1*cos(x1(i))==r2+r3*cos(x2)+r4*cos(x3);
    eq2=r1*sin(x1(i))==r5+r3*sin(x2)+r4*sin(x3);
    sol(i)=vpasolve([eq1 eq2],[x2 x3]);
end

The solutions are stored in sol variable. You can reach them by typing

sol(1).x2
sol(1).x3

sol(2).x2
sol(2).x3

.
.

and so on.

[Kubilay AKPINAR]

This gives the answes in radians right? If I want to work on degree I must change cos or sin codes with cosd sind, am I correct?

[Birdman]

Yes, exactly. Or simply you can multiply the outcomes with 180 and then divide them to pi.

[Kubilay AKPINAR]

Thanks for your reply, but I don't understand how this code works. For example, how many values does x2 have? Because it yields different values when I enter sol(1).x2 or sol(2)x2. Could you please clarify that?

[Birdman]

The changing values of x2 and x3 is dependent on x1. For instance, when x1=100, you have a pair of x2 and x3. When x1=101, you have another pair of x2 and x3 and so forth.

[Kubilay AKPINAR]

Thank you very much, I have one more question, are these anwers correct? I mean, are they approximate answers or exact anwers? I am asking that because there are 4 unknowns (cosx2 sinx2 cosx3 sinx3) but 2 equations. According to mathematical rules you can not solve this problem in a normal way.

[Birdman]

No, there are two unknowns and two equations. You can solve automatically. If it had been as you said, then MATLAB could not have solved this pair of equations. cos and sin are functions, do not get confused.

[Kubilay AKPINAR]

Okey, thank you for your explaination and effort.

[Birdman]

You are welcome.

[Kubilay AKPINAR]

Hello again,

I found a chance to try your codes with the known values (for example, r1=7.25 etc.) but I am having an error.

Here it is the code;

r1=7.25; r2=1.85; r3=5.79; r4=1.43; r5=3.45; %random values
x1=100:1:126;
syms x2 x3
for i=1:numel(x1)
    eq1=r1*cos(x1(i))==r2+r3*cos(x2)+r4*cos(x3);
    eq2=r1*sin(x1(i))==r5+r3*sin(x2)+r4*sin(x3);
    sol(i)=vpasolve([eq1 eq2],[x2 x3]);
end

And this is the answer;

>> sol(1).x2
 
ans =
 
Empty sym: 0-by-1
 
>> sol(1)

ans = 

  struct with fields:

    x2: [0×1 sym]
    x3: [0×1 sym]

>> 

Could you please help me?

[Birdman]

Hello,

If there is no solution found, then it means for r1=7.25 and x1=100, there is no numerical solution. Instead, you can try symbolic approach with solve command as follows:

r1=7.25; r2=1.85; r3=5.79; r4=1.43; r5=3.45; %random values
x1=100:1:126;
syms x2 x3
for i=1:numel(x1)
    eq1=r1*cos(x1(i))==r2+r3*cos(x2)+r4*cos(x3);
    eq2=r1*sin(x1(i))==r5+r3*sin(x2)+r4*sin(x3);
    sol(i)=solve([eq1 eq2],[x2 x3]);
end

Then, by typing

vpa(sol(1).x2)
vpa(sol(1).x3)
.
.

and the same for the rest of the solutions, you can obtain the results. The results may be in imaginary form.

[Kubilay AKPINAR]

Hello again,

I am sorry for late response but your codes work perfectly, it was my fault, I have correct answers now. However, I am continuing to solve other problems just like the one I posted here by using your way but I am having another problem.

This is my code;

format short g
%Angle
ro2b=7.25; ro2d=1.85; ro4a=5.79; rab=1.43; ro4d=3.45;
th2=100:126;
syms x2 x3
th4=zeros(1,27);
th3=zeros(1,27);
for i=1:numel(th2)
    eq1=ro2b*cosd(th2(i))==ro2d+ro4a*cosd(x2)+rab*cosd(x3);
    eq4=ro2b*sind(th2(i))==ro4d+ro4a*sind(x2)+rab*sind(x3);
    sol(i)=vpasolve([eq1 eq4],[x2 x3]);
end

for j=1:27
    th4(1,j)=sol(j).x2;
    th3(1,j)=sol(j).x3;
end

%Velocity
w2=4;
syms w3 w4
for t=1:numel(th2)
    eq3=-ro2b*sind(th2(t))*w2==-ro4a*sind(th4(t,1))*w4-rab*sind(th3(t,1))*w3;
    eq4=ro2b*cosd(th2(t))*w2==ro4a*cosd(th4(t,1))*w4+rab*cosd(th3(t,1))*w3;
    sol(t)=vpasolve([eq3 eq4],[w3 w4]);%PROBLEM IS HERE
end

w3=zeros(27,1);
w4=zeros(27,1);
for a=1:27
    w3(a,1)=sol(a).w3;
    w4(a,1)=sol(a).w4;
end

I get correct results for theta2 theta3 and theta4 but when I try to solve another equation by using the same way I am having this error;

Subscripted assignment between dissimilar structures.

Error in Untitled2 (line 26)
    sol(t)=vpasolve([eq3 eq4],[w3 w4]);

I did a search online but I couldn't understand answers, I think there's something different. Could you help me please?

[Kubilay AKPINAR]

I detected my mistake, I must use different names for this piece of code;

sol(i)=vpasolve([eq1 eq4],[x2 x3]);

sol(t)=vpasolve([eq3 eq4],[w3 w4]);

there are two "sol" matrices, that's the problem. I changed the second one as "solu" and the error disappeared. Thanks anyway.