MATLAB: Solve the following simultaneous set of nonlinear trigonometric equations:

Question

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.

Answer 1

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