Playing with robots for kids - Which letter will I reach?
JavaScript (ES6), 194 176 169 163 bytes
Saved some bytes thanks to @Luke and @Arnauld.
s=&gt;(p=35,d=3,t='ERLBIXN1OAQYCTG1FWHPDZS11KVUMJ11111 11',[...s].every(i=&gt;i=='L'?d--:i&gt;'Q'?d++:i&lt;'G'?+t[p+=[1,8,-1,-8][d%4]]||!t[p]?p=1/0:p:0)?'':t[p]||'Out of mat')

Ungolfed:
s=&gt;(
  p=35,
  d=3,
  t='ERLBIXN1OAQYCTG1FWHPDZS11KVUMJ11111 11',
  [...s].every(i=&gt;i=='L'?d--:
                  i&lt;'Q'?d++:
                  i&lt;'G'?+t[p+=[1,8,-1,-8][d%4]]||!t[p]?p=1/0:p:
                  0
              )?'':
               t[p]||'Out of mat'
)



f=
s=&gt;(p=35,d=3,t='ERLBIXN1OAQYCTG1FWHPDZS11KVUMJ11111 11',[...s].every(i=&gt;i=='L'?d--:i&gt;'Q'?d++:i&lt;'G'?+t[p+=[1,8,-1,-8][d%4]]||!t[p]?p=1/0:p:0)?'':t[p]||'Out of mat')

console.log(f('FFG')); //P
console.log(f('FRFRFG')); //Out of mat
console.log(f('RRFFG')); //Out of mat
console.log(f('FFFRFFLFG')); //X
console.log(f('FFFRFFLF')); //(space)
console.log(f('FFFRRFFFG')); //(space)
console.log(f('RRRRRLFFFLFFRFRFGFFRRGRFF')); //L
console.log(f('FFFFFRRFG')); //Out of mat