Solve the following simultaneous set of nonlinear trigonometric equations:

I want to solve the given 12 equations for finding the joint angles of a manipulator. Is there anyway in matlab to solve them? Here are the equations and code:

syms c1 c2 c3 c4 c5 c6 s1 s2 s3 s4 s5 s6 d2 d3 d6 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12
e1=c1*[c2*(c4*c5*c6 - s4*s6) -s2*s5*c6] - s1*(s4*c5*c6 + c4*s6) == 0
e2=s1*[c2*(c4*c5*c6 - s4*s6) - s2*s5*c6] + c1*(s4*c5*c6 + c4*s6) == 0
e3=-s2*(c4*c5*c6- s4*s6)- c2*s5*c6 == 1
e4=c1*[-c2*(c4*c5*s6 + s4*c6) + s2*s5*s6] - s1*(-s4*c5*s6 + c4*c6) == 1
e5=s1*[-c2*(c4*c5*s6 + s4*c6) + s2*s5*s6] + c1*(-s4*c5*s6 + c4*c6) == 0
e6=s2*(c4*c5*s6 + s4*c6) + c2*s5*s6 == 0
e7=c1*(c2*c4*s5 + s2*c5) - s1*s4*s5 == 0
e8=s1*(c2*c4*s5 + s2*c5) + c1*s4*s5 == 1
e9=-s2*c4*s5 + c2*c5 == 0
e10=c1*s2*d3 - s1*0154 + .263*(c1*c2*c4*s5 + c1*c5*s2 -s1*s4*s5) == -0.154
e11=s1*s2*d3 + c1*0.154 + .263*(c1*s4*s5 + c2*c4*s1*s5 + c5*s1*s2) == 0.763
e12=c2*d3 + .263*(c2*c5 - c4*s2*s5) == 0
d6==.253
d2==0.154

I tried using solve functions like this:

sol = solve([e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12], [c1, c2, c3, c4, c5, c6, s1, s2, s3, s4, s5, s6, d3]);

But it does not work.

How can I find the values of c1,s1......?

Thank you.

[David Hill]

Seems to work just find for me.

[Robert]

I am trying to achieve a single value

[David Hill]

Pick a solution you want (1 of 28);

[Robert]

Got it... Thanks..

Answer 1

[Walter Roberson]

c1 = 0, c2 = 0, c4 = 0, c5 = -1000, c6 = 0, d3 = -500, s1 = 1/1000, s2 = -1, s4 = -1/s6, s5 = 0, s6 = s6

c1 = 0, c2 = 0, c4 = 0, c5 = -1000, c6 = 0, d3 = -500, s1 = 1/1000, s2 = -1, s4 = -1/s6, s5 = 0, s6 = s6

c1 = 0, c2 = 0, c4 = 0, c5 = 1000, c6 = 0, d3 = 500, s1 = 1/1000, s2 = 1, s4 = 1/s6, s5 = 0, s6 = s6

c1 = 0, c2 = 0, c4 = 0, c5 = 1000, c6 = 0, d3 = 500, s1 = 1/1000, s2 = 1,  s4 = 1/s6, s5 = 0, s6 = s6

c1 = 250/77, c2 = -1, c4 = -5929000/62500005929, c5 = 0, c6 = 1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = 19250000000/62500005929, s5 = 1, s6 = 0

c1 = 250/77, c2 = -1, c4 = -5929000/62500005929, c5 = 0, c6 = 1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = 19250000000/62500005929, s5 = 1, s6 = 0

c1 = 250/77, c2 = -1, c4 = 5929000/62500005929, c5 = 0, c6 = -1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = -19250000000/62500005929, s5 = -1, s6 = 0

c1 = 250/77, c2 = -1, c4 = 5929000/62500005929, c5 = 0, c6 = -1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = -19250000000/62500005929, s5 = -1, s6 = 0

c1 = 250/77, c2 = 0, c4 = 0, c5 = 0, c6 = 0, d3 = -5929/125000*s6, s1 = 0, s2 = 1/s6, s4 = 1, s5 = 77/250, s6 = s6

c1 = 250/77, c2 = 0, c4 = 0, c5 = 0, c6 = 0, d3 = -5929/125000*s6, s1 = 0, s2 = 1/s6, s4 = 1, s5 = 77/250, s6 = s6

c1 = 250/77, c2 = 0, c4 = 0, c5 = 0, c6 = 0, d3 = 5929/125000*s6, s1 = 0, s2 = -1/s6, s4 = -1, s5 = -77/250, s6 = s6

c1 = 250/77, c2 = 0, c4 = 0, c5 = 0, c6 = 0, d3 = 5929/125000*s6, s1 = 0, s2 = -1/s6, s4 = -1, s5 = -77/250, s6 = s6

c1 = 250/77, c2 = 1, c4 = -5929000/62500005929, c5 = 0, c6 = 1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = -19250000000/62500005929, s5 = -1, s6 = 0

c1 = 250/77, c2 = 1, c4 = -5929000/62500005929, c5 = 0, c6 = 1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = -19250000000/62500005929, s5 = -1, s6 = 0

c1 = 250/77, c2 = 1, c4 = 5929000/62500005929, c5 = 0, c6 = -1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = 19250000000/62500005929, s5 = 1, s6 = 0

c1 = 250/77, c2 = 1, c4 = 5929000/62500005929, c5 = 0, c6 = -1, d3 = 0, s1 = 1/1000, s2 = 0, s4 = 19250000000/62500005929, s5 = 1, s6 = 0

c1 = 125/77-1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = -5929* s1/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), c6 = 0, d3 = 1/500*(125+(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = 1, s4 = -1, s5 = (9625-77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), s6 = -1

c1 = 125/77-1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = -5929*s1/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), c6 = 0, d3 = 1/500*(125+(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = 1, s4 = 1, s5 = (-9625+77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), s6 = 1

c1 = 125/77-1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = 5929*s1/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), c6 = 0, d3 = 1/500*(-125-(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = -1, s4 = -1, s5 = (9625-77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), s6 = 1

c1 = 125/77-1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = 5929*s1/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), c6 = 0, d3 = 1/500*(-125-(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = -1, s4 = 1, s5 = (-9625+77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)-5929*s1-31250), s6 = -1

c1 = 125/77+1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = -5929*s1/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), c6 = 0, d3 = 1/500*(-125+(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = -1, s4 = -1, s5 = (-9625-77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), s6 = 1

c1 = 125/77+1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = -5929*s1/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), c6 = 0, d3 = 1/500*(-125+(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = -1, s4 = 1, s5 = (9625+77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), s6 = -1

c1 = 125/77+1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = 5929*s1/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), c6 = 0, d3 = 1/500*(125-(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = 1, s4 = -1, s5 = (-9625-77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), s6 = -1

c1 = 125/77+1/77*(-5929000*s1^2+5929*s1+15625)^(1/2), c2 = 0, c4 = 0, c5 = 5929*s1/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), c6 = 0, d3 = 1/500*(125-(-5929000*s1^2+5929*s1+15625)^(1/2))/s1, s1 = s1, s2 = 1, s4 = 1, s5 = (9625+77*(-5929000*s1^2+5929*s1+15625)^(1/2))/(-5923071*s1^2+250*(-5929000*s1^2+5929*s1+15625)^(1/2)+5929*s1+31250), s6 = 1

[Robert]

Thank you for your reply. I also had this kind of result but is it possible to attain a single value for each variable?

[Robert]

And also could you please tell me to show all the values at once?

[Walter Roberson]

sol = solve([e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12], [c1, c2, c3, c4, c5, c6, s1, s2, s3, s4, s5, s6, d3]);
sols = [sol.c1, sol.c2, sol.c3, sol.c4, sol.c5, sol.c6, sol.s1, sol.s2, sol.s3, sol.s4, sol.s5, sol.s6, sol.d3]);

[Walter Roberson]

Do you have restrictions such as particular variables having to be positive?

[Robert]

No. I do not have any restrictions

[Walter Roberson]

Then pick one of the many solutions at random, unless you have some guidance as to why one solution might be better than another.

[Robert]

Got it... thanks a lot

[Robert]

[Robert]

Hi, I have got something like this when I used the code... could you please share me your code?

[Walter Roberson]

eqns := [-s1*(c5*c6*s4 + c4*s6) - c1*c2*(-c4*c5*c6 + s4*s6), -s1*(c5*c6*s4 + c4*s6) - c1*c6*s2*s5, c1*(c5*c6*s4 + c4*s6) - s1*(c2*(-c4*c5*c6 + s4*s6) + c6*s2*s5), s2*(-c4*c5*c6 + s4*s6) - c2*c6*s5 = 1, -c1*(c2*(c4*c5*s6 + c6*s4) - s2*s5*s6) - s1*(-c5*s4*s6 + c4*c6) = 1, c1*(-c5*s4*s6 + c4*c6) - s1*(c2*(c4*c5*s6 + c6*s4) - s2*s5*s6), s2*(c4*c5*s6 + c6*s4) + c2*s5*s6, c1*(c2*c4*s5 + c5*s2) - s1*s4*s5, s1*(c2*c4*s5 + c5*s2) + c1*s4*s5 = 1, -c4*s2*s5 + c2*c5, (263*c1*c5*s2)/1000 - 154*s1 + c1*d3*s2 - (263*s1*s4*s5)/1000 + (263*c1*c2*c4*s5)/1000 = -77/500, (77*c1)/500 + (263*c5*s1*s2)/1000 + (263*c1*s4*s5)/1000 + d3*s1*s2 + (263*c2*c4*s1*s5)/1000 = 763/1000, (263*c2*c5)/1000 + c2*d3 - (263*c4*s2*s5)/1000];
[solve](eqns);
[allvalues](%);
ListTools:-Flatten(%);
sort(%);
lprint(%);

[Robert]

Thank you for sharing your code. But when I run this code in matlab, it shows a bunch of error starting from the start. Could you please check it again?

[Walter Roberson]

You asked for my code and I provided it. I never said it was MATLAB code.

Just take the sols that you got from MATLAB and choose a row at random -- unless you have reason to select one form over another. For example at least 8 of the solutions you got involve complex values, and perhaps you prefer real-only solutions even though you specifically told us there were no restrictions on the variables.

There is not just one solution to the equations you provided. Every solution is a member of a pair involving the solution of a quadratic, so there are an even number of solutions in pairs.

[Robert]

Thank you for your comment.

I understand. And yes, I do not have any restrictions on choosing the values and I prefer choosing the real values as it makes the process easier. I know that it can have more than one solution as it is for inverse kinematics of a robotic arm.

Excuse me but I am fairly new to matlab that is why I do not have much knowledge in coding. I just wanted to have output like yours as it is much easier to comprehend than mine. That is why I was asking for your matlab code. I thought you did it in Matlab.

[Walter Roberson]

syms c1 c2 c3 c4 c5 c6 s1 s2 s3 s4 s5 s6 d2 d3 d6 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12
e1=c1*[c2*(c4*c5*c6 - s4*s6) -s2*s5*c6] - s1*(s4*c5*c6 + c4*s6) == 0;
e2=s1*[c2*(c4*c5*c6 - s4*s6) - s2*s5*c6] + c1*(s4*c5*c6 + c4*s6) == 0;
e3=-s2*(c4*c5*c6- s4*s6)- c2*s5*c6 == 1;
e4=c1*[-c2*(c4*c5*s6 + s4*c6) + s2*s5*s6] - s1*(-s4*c5*s6 + c4*c6) == 1;
e5=s1*[-c2*(c4*c5*s6 + s4*c6) + s2*s5*s6] + c1*(-s4*c5*s6 + c4*c6) == 0;
e6=s2*(c4*c5*s6 + s4*c6) + c2*s5*s6 == 0;
e7=c1*(c2*c4*s5 + s2*c5) - s1*s4*s5 == 0;
e8=s1*(c2*c4*s5 + s2*c5) + c1*s4*s5 == 1;
e9=-s2*c4*s5 + c2*c5 == 0;
e10=c1*s2*d3 - s1*0154 + .263*(c1*c2*c4*s5 + c1*c5*s2 -s1*s4*s5) == -0.154;
e11=s1*s2*d3 + c1*0.154 + .263*(c1*s4*s5 + c2*c4*s1*s5 + c5*s1*s2) == 0.763;
e12=c2*d3 + .263*(c2*c5 - c4*s2*s5) == 0;
%d6==.253;
%d2==0.154;

sol = solve([e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12], [c1, c2, c3, c4, c5, c6, s1, s2, s3, s4, s5, s6, d3]);
sols = array2table(double([sol.c1, sol.c2, sol.c3, sol.c4, sol.c5, sol.c6, sol.s1, sol.s2, sol.s3, sol.s4, sol.s5, sol.s6, sol.d3]), 'VariableNames', string([c1, c2, c3, c4, c5, c6, s1, s2, s3, s4, s5, s6, d3]))

sols = 28x13 table
      c1      c2    c3        c4             c5         c6     s1         s2        s3      s4        s5        s6          d3    
    ______    __    __    ___________    ___________    __    _____    _________    __    ______    ______    _______    _________

    3.2468     0    0               0              0     0        0            1    0         -1    -0.308         -1    -0.047432
    3.2468     0    0               0              0     0        0            1    0          1     0.308          1    -0.047432
    3.2468     0    0               0              0     0        0           -1    0          1     0.308         -1     0.047432
    3.2468     0    0               0              0     0        0           -1    0         -1    -0.308          1     0.047432
         0     0    0               0           1000     0    0.001            1    0          1         0          1          500
    3.2468     0    0               0              0     0        0    -0.047432    0          1     0.308    -21.083            1
    3.2468     0    0               0              0     0        0    -0.047432    0         -1    -0.308     21.083            1
    3.2468    -1    0      9.4864e-05              0    -1    0.001            0    0     -0.308        -1          0            0
    3.2468     1    0     -9.4864e-05              0     1    0.001            0    0     -0.308        -1          0            0
    3.2468     1    0      9.4864e-05              0    -1    0.001            0    0      0.308         1          0            0
    3.2468    -1    0     -9.4864e-05              0     1    0.001            0    0      0.308         1          0            0
    3.2468     0    0               0     9.4864e-05     0    0.001            1    0         -1    -0.308         -1            0
    3.2468     0    0               0    -9.4864e-05     0    0.001           -1    0          1     0.308         -1            0
    3.2468     0    0               0    -9.4864e-05     0    0.001           -1    0         -1    -0.308          1            0
    3.2468     0    0               0     9.4864e-05     0    0.001            1    0          1     0.308          1            0
         0     0    0               0          -1000     0    0.001           -1    0         -1         0          1         -500

[Robert]

Thank you very much for your help. I really appreciate it.