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Equation with two unknowns in complex exponential

  1. Oct 27, 2013 #1
    hello friends,
    when i build the mathmatical model of robot,i face a new question that i ever seen before.
    i have a reverse kinematic lever as the leg and i want to use the tip position to get the relationship of fold angle and rotate angle reversely
    here is my equation:
    x*e^iθ - y*e^iθ * e^-iα + z*e^iθ = d
    x is the known length of first section of kinematic lever
    y is the known length of second section of kinematic lever
    z is the known length of third section of kinematic lever
    d is the known length from {0,0} (which is the pin point of kinematic lever rotation point) to kinematic lever tip
    i want to find the exact relation of θ and α

    could you help me to get the result of the equation? thanks.
  2. jcsd
  3. Oct 28, 2013 #2


    User Avatar

    Staff: Mentor

    One equation and two unknowns. The problem is underdefined and there is more than one solution.
  4. Oct 28, 2013 #3
    If I'm parsing your equation correctly, it can be rearranged to say [tex]e^{i\theta}= \frac{d}{x+z - ye^{-i\alpha}}[/tex] which pins down [itex]\theta[/itex] (modulo [itex]2\pi[/itex]) for any given value of [itex]\alpha[/itex].
  5. Oct 30, 2013 #4
    sorry,i made a mistake,the correct equation should be :
    x*e^iθ - y*e^-iθ * e^-iα + z*e^iθ = d*e^iβ
    d*e^iβ is the complex number of pin to tip vector,which β is known.
    to DrClaude,
    yes it may be have two answer,i just need the exact relation of θ and α.
  6. Oct 30, 2013 #5
    to economicsnerd,
    thanks for your answer, but it would go further,and get the f(θ) = α
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