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#Mathematica 92 49
Calle deserves full credit for streamlining the code.
f@l_:=2 N@Norm[Tr[l/.{"r"→1,"l"→-1,"u"→I,"d"→-I}]]
Example
f[{"u", "u", "u", "u", "d", "r", "r", "r", "l"}]
7.2111
#Mathematica 9292 49
f@l_ := 2 √Tr[{Re@#, Im@#}²] &@Tr[Select[ToLowerCase@l =2 N@Norm[Tr[l/. {"r" -> 1"r"→1, "l" -> "l"→-1, "u" -> I"u"→I, "d" -> "d"→-I}, NumericQ]] // N]]
f[{"u", "U""u", "u", "u", "d","X", "r", "r", "r", "l"}]
#Mathematica 92
f@l_ := 2 √Tr[{Re@#, Im@#}²] &@Tr[Select[ToLowerCase@l /. {"r" -> 1, "l" -> -1, "u" -> I, "d" -> -I}, NumericQ]] // N
f[{"u", "U", "u", "u", "d","X", "r", "r", "r", "l"}]
