Efficient Robot Movement

Retina, 103 74 71 bytes

<
>>>
+`(>(\^*>){3})\^
^$1
+`\^(>\^*>)\^
$1
>>(\^*)>(\^+)
<$2<$1
<?>*$

Try it online! Link includes test cases. Explanation:

<
>>>

Turn left turns into triple right turns.

+`(>(\^*>){3})\^
^$1

Reduce all turns modulo 4.

+`\^(>\^*>)\^
$1

Cancel out movements in opposite directions.

>>(\^*)>(\^+)
<$2<$1

Turn a triple right turn back into a left turn. This also handles the case of >>^>^ which needs to become <^<^.

<?>*$

Delete unnecessary trailing turns.

Mathematica, 135 bytes

{a="^"~Table~Ramp@#&;a@#,s=If[#2>0,">","<"],a@Abs@#2,If[#<0,s,""],a@-#}<>""&@@ReIm[j=0;i=1;Switch[#,">",i*=I,"<",i/=I,"^",j+=i]&/@#;j]&

Takes a List of strings as input.

Explanation

j=0;i=1

Set j to 0, and set i to 1.

/@#

For each character in input...

Switch[#,">",i*=I,"<",i/=I,"^",j+=i]

If the character is >, multiply i by the imaginary unit. If the character is >, divide i by the imaginary unit. If the character is ^, add i to j.

ReIm[ ... ;j]

Take the real and imaginary parts of j. This gives the Cartesian coordinate of the robot.

... &@@

Apply the following to this result:

---

a="^"~Table~Ramp@#&;

Set a to a function that generates a string with (input) or 0 character ^s, whichever is larger.

{ ... }

A List consisting of...

a@#

a applied to the first input (real part of j)

s=If[#2>0,">","<"]

If the second input (imaginary part of j) is larger than 0, >. Otherwise, <. Set s to the resulting character.

a@Abs@#2

a applied to the absolute value of the second input.

If[#<0,s,""]

If the first input is less than 0, s. Otherwise, empty string.

a@-#

Apply a to the input times negative one.

... <>""

Join the strings.

Mathematica 119 Bytes

JungHwan's final position to path code was shorter than mine, so using that. I think there is probably an even shorter way to do this...

I use the built-in AnglePath function to decide the final position. I also define the symbols L, F, and R for "<", "^" and ">", to save a few quote characters.

L={0,Pi/2};R=-L;F={1,0};{a="F"~Table~Ramp@#&;a@#,s=If[#2>0,"L","R"],a@Abs@#2,If[#<0,s,""],a@-#}<>""&@@Last@AnglePath@#&

Usage:

%@{R,F,L,L,F,F,R,F,F}

Output:

FFLF