This comes from chapter 6 of Ideals, Varieties and Algorithms by Cox et al.
The equations are from a planar robot with three joints and one prismatic joint. See the following picture:
Given a hand position $(a,b)$ and orientation $(x,y)$ where $x=\cos \alpha, y=\sin \alpha$, for some angle $\alpha$, we set up the following equations: $$l4(c1(c2c3-s2s3)-s1(c2s3+c3s2))+l3(c1c2-s1s2)+l2c1=a\\ l4(s1(c2c3-s2s3)-c1(c2s3+c3s2))+l3(s1c2+s2c1)+l2s1=b\\ c1^2+s1^2=1\\ c2^2+s2^2=1\\ c3^2+s3^2=1\\ c1(c2c3-s2s3)-s1(c2s3+c3s2)=x\\ s1(c2c3-s2s3)-c1(c2s3+c3s2)=y$$
where $\cos\theta_i=ci, \sin\theta_i=si$.
I am trying to find a Groebner basis when $l2=l3=1$ using Mupad. The variables are $l4, ci, si$. I tried several lex orders. None of them gives a result.
Any suggestions? Thank you for any help!