Solving trignometric simultaneous equations

[peteculmer]

Solving trignometric simultaneous equations...

Hi,

I'm in need of some help solving a set of simultaneous equations which will (hopefully!) solve the inverse kinematics for a robot arm. The equations are as follows:

eq1 := Xu-Xl = -c[4]*L_u2-s[4]*c[5]*L_l*c[6]-c[4]*L_l*s[6]+c[4]*L_u1

eq2 := Zu-Zl = c[3]*s[4]*L_u2-(c[3]*c[4]*c[5]-s[3]*s[5])*L_l*c[6]+c[3]*s[4]*L_l*s[6]-c[3]*s[4]*L_u1

eq3 := Yl = (s[3]*c[4]*c[5]+c[3]*s[5])*L_l*c[6]-Yu*(-L_u1+L_l*s[6])/L_u2

where s[3] denotes sin(theta3) and c[3] cos[theta3] etc

The unknowns I am trying to solve for are theta3,theta4 and theta5.
I have been attempting to use the solve command in maple but to no effect. If anybody has any pointers of what I could try,either methods or different packages etc I would be very grateful,
Thanks
Pete Culmer

EDIT:

I should point out that all other variables in the equations are known. I'm fairly sure there should be one or more symbolic solution(s)...hmmmm

 

[dextercioby]

Advice:drope it dead.These are transcendental equations.Even if u'd be expressing everything in terms of "sines" (par éxample),it would be a system of irrational equations (involving radicals from unknowns).And besides,the number of equations is much less than the number of unknowns.

Daniel.

 

[tacman]

Hi Pete,

Solving simultaneous equations involving trigonometric functions can be challenging, but there are a few strategies you can try. One approach is to use the elimination method, where you eliminate one variable at a time by substituting one equation into another. In your case, you could start by solving for theta3 in eq3 and substituting that into eq1 and eq2. This will eliminate theta3 from those equations, leaving you with two equations and two unknowns (theta4 and theta5). From there, you can use substitution or elimination again to solve for those remaining variables.

Another approach is to use the trigonometric identities to simplify the equations before attempting to solve them. For example, you could use the double angle formula to rewrite c[4]^2 and s[4]^2 in terms of c[8] and s[8], where c[8] denotes cos(2*theta4) and s[8] denotes sin(2*theta4). This may make the equations easier to solve.

You could also try using a different software package, such as MATLAB or Mathematica, to solve the equations. Each package has its own strengths and may have a more efficient way of solving your specific set of equations.

I hope these suggestions help and good luck with solving your equations!