//---Robot dell'esempio 1, [CLO, p. 273]
ring r=(0,l3,l2,a,b),(c2,s2,c1,s1),lp;
option(redSB);
ideal id=l3*(c1*c2-s1*s2)+l2*c1-a,
l3*(c1*s2+c2*s1)+l2*s1-b,
c1^2+s1^2-1,c2^2+s2^2-1;
ideal j=std(id);
//--- base di Groebner dell'ideale id
j;
j[1]=(4*l2^2*a^2+4*l2^2*b^2)*s1^2+(4*l3^2*l2*b-4*l2^3*b-4*l2*a^2*b-4*l2*b^3)*s1+(l3^4-2*l3^2*l2^2-2*l3^2*a^2-2*l3^2*b^2+l2^4-2*l2^2*a^2+2*l2^2*b^2+a^4+2*a^2*b^2+b^4)
j[2]=(2*l2*a)*c1+(2*l2*b)*s1+(l3^2-l2^2-a^2-b^2)
j[3]=(2*l3*l2*a)*s2+(2*l2*a^2+2*l2*b^2)*s1+(l3^2*b-l2^2*b-a^2*b-b^3)
j[4]=(2*l3*l2)*c2+(l3^2+l2^2-a^2-b^2)
//---Pomiamo l2 = l3 =1
ideal ids=subst(id,l3,1,l2,1);
ids;
ids[1]=c2*c1-s2*s1+c1+(-a)
ids[2]=c2*s1+s2*c1+s1+(-b)
ids[3]=c1^2+s1^2-1
ids[4]=c2^2+s2^2-1
//--- base di Groebner dell'ideale ids
ideal gids = std(ids);
gids;
gids[1]=(4*a^2+4*b^2)*s1^2+(-4*a^2*b-4*b^3)*s1+(a^4+2*a^2*b^2-4*a^2+b^4)
gids[2]=(2*a)*c1+(2*b)*s1+(-a^2-b^2)
gids[3]=(2*a)*s2+(2*a^2+2*b^2)*s1+(-a^2*b-b^3)
gids[4]=2*c2+(-a^2-b^2+2)
//--- per discutere il caso a = 0 bisogna ricalcolare
//    la base di Groebner
ideal ida0 = subst(ids,a,0);
ida0;
ida0[1]=c2*c1-s2*s1+c1
ida0[2]=c2*s1+s2*c1+s1+(-b)
ida0[3]=c1^2+s1^2-1
ida0[4]=c2^2+s2^2-1
std(ida0);
_[1]=2*s1+(-b)
_[2]=4*c1^2+(b^2-4)
_[3]=s2+(-b)*c1
_[4]=2*c2+(-b^2+2)
//---Risolviamo il problema cinematico inverso,
//   cioe' dato il posizionamento della mano
//   (a,b) calcoliamo (c2,s2,c1,s1)=: elenco l.
LIB "solve.lib";
// ** loaded /usr/bin/../share/singular/LIB/solve.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/ring.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/primdec.lib (4.0.1.2,Jan_2015)
// ** loaded /usr/bin/../share/singular/LIB/absfact.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/matrix.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/nctools.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/random.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/poly.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/elim.lib (4.0.0.1,Jan_2014)
// ** loaded /usr/bin/../share/singular/LIB/general.lib (4.0.0.1,Jan_2014)
// ** loaded /usr/bin/../share/singular/LIB/inout.lib (4.0.0.0,Jun_2013)
// ** loaded /usr/bin/../share/singular/LIB/triang.lib (4.0.0.0,Jun_2013)
list l=solve(ids,6,0);
ring re=(real,6),(c2,s2,c1,s1),lp;
ideal reids = imap(r,ids);
//--- bisogna fare il passaggio in due passi
ring re=(real,6),(c2,s2,c1,s1,l3,l2,a,b),lp;
// ** redefining re **
ideal re_ids = imap(r,ids);
ids;
setring r;
ids;
ids[1]=c2*c1-s2*s1+c1+(-a)
ids[2]=c2*s1+s2*c1+s1+(-b)
ids[3]=c1^2+s1^2-1
ids[4]=c2^2+s2^2-1
ring re=(real,6),(c2,s2,c1,s1,l3,l2,a,b),lp;
// ** redefining re **
ideal re_ids = imap(r,ids);
ideal id2 = c2*c1-s2*s1+c1+(-a), c2*s1+s2*c1+s1+(-b),=c1^2+s1^2-1,c2^2+s2^2-1;
   skipping text from `c1`
ideal id2 = c2*c1-s2*s1+c1+(-a), c2*s1+s2*c1+s1+(-b),c1^2+s1^2-1,c2^2+s2^2-1;
ring re2=(real,6),(c2,s2,c1,s1,a,b),lp;
ideal id3 = imap(re,id2);
ideal id4 = a-1, b-1;
ideal id5 = id3 + id4;
list l=solve(id5,6,0);
// ** redefining l **
[1]:
   [1]:
      0
   [2]:
      1
   [3]:
      1
   [4]:
      0
   [5]:
      1
   [6]:
      1
[2]:
   [1]:
      0
   [2]:
      -1
   [3]:
      0
   [4]:
      1
   [5]:
      1
   [6]:
      1

// 'solve' created a ring, in which a list SOL of numbers (the complex solutions)
// is stored.
// To access the list of complex solutions, type (if the name R was assigned
// to the return value):
        setring R; SOL; 


//---Robot dell'esempio 2, [CLO, p. 274]

ring r=(0,l4,l3,l2,a,b,c,s),(c2,s2,c1,s1),lp;
// ** redefining r **
option(redSB);
ideal id=l4*c+l3*(c1*c2-s1*s2)+l2*c1-a,
l4*s+l3*(c1*s2+c2*s1)+l2*s1-b,
c1^2+s1^2-1,c2^2+s2^2-1;
id;
id[1]=(l3)*c2*c1+(-l3)*s2*s1+(l2)*c1+(l4*c-a)
id[2]=(l3)*c2*s1+(l3)*s2*c1+(l2)*s1+(l4*s-b)
id[3]=c1^2+s1^2-1
id[4]=c2^2+s2^2-1
monitor("");