How to solve the triangulation problem?I have 3 sensors. I've built the following system of equations that match the data from sensors. I need to find x, y, R a , R b , R c , α, β and γ. Can you please help me, or at least hint how can I solve this system. A x = x + R a cos ⁡ α A y = y + R a sin ⁡ α B x = x + R b cos ⁡ β B y = y + R b sin ⁡ β C x = x + R c cos ⁡ γ C y = y + R c sin ⁡ γ R b − R a = z 1 R c − R a = z 2

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How to solve the triangulation problem?I have 3 sensors. I&#039;ve built the following system of equations that match the data from sensors. I need to find  x,   y,       R    a  ,       R    b  ,       R    c  ,   α,   β and   γ. Can you please help me, or at least hint how can I solve this system.                              A          x                                    =        x        +                  R          a                cos        ⁡        α                                      A          y                                    =        y        +                  R          a                sin        ⁡        α                                      B          x                                    =        x        +                  R          b                cos        ⁡        β                                      B          y                                    =        y        +                  R          b                sin        ⁡        β                                      C          x                                    =        x        +                  R          c                cos        ⁡        γ                                      C          y                                    =        y        +                  R          c                sin        ⁡        γ                                      R          b                −                  R          a                                    =                  z          1                                              R          c                −                  R          a                                    =                  z          2

megagoalai · Accepted Answer

Hint: start with       R    b    =      z    1    +      R    a  . Then by rearranging the first two equations, squaring and adding, we can obtain:       R    a    2    =  (  A  −  1      )    2    (      x    2    +      y    2    ). Similarly, squaring and adding third and fourth equation, we get   (      R    a    +      z    1        )    2    =  (  B  −  1      )    2    (      x    2    +      y    2    ).Thus,            (                        R          a                                      R            a                    +                      z            1                              )        2    =            (                        A          −          1                          B          −          1                    )        2

Deon Moran · Accepted Answer

From a trigonometric identity, the values of x and y must satisfy these equations simultaneously:      x    2    +      y    2    =            R      a      2              (      A      −      1              )        2                  x    2    +      y    2    =            R      b      2              (      B      −      1              )        2                  x    2    +      y    2    =            R      c      2              (      C      −      1              )        2              .The only way for three circles with the same center (in this case, the origin) to have any real intersection points is for the circles to be identical: that is:            R      a      2              (      A      −      1              )        2              =            R      b      2              (      B      −      1              )        2              =            R      c      2              (      C      −      1              )        2              .From the second to last equation,      |                  B        −        1                    A        −        1              |        R    a    −      R    a    =      z    1      ⟹        R    a    =            z      1                      |                              B            −            1                                A            −            1                          |            −      1        .Therefore,      R    b    =      z    1    +            z      1                      |                              B            −            1                                A            −            1                          |            −      1        ,        R    c    =      z    2    +            z      1                      |                              B            −            1                                A            −            1                          |            −      1        .This should give you a good start.As far as finding the angles   α  ,  β  ,  γ, I&#039;m not sure at the moment. Perhaps someone else can shed light on this.

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