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Serial link chain manipulator with constrained geometry

  1. Nov 23, 2011 #1
    1. The problem statement, all variables and given/known data

    I have a three link revolute manipulator at the origin. I know all the link lengths. The joint angle for the third link is coupled to the second such that Theta3 = k*Theta2. I want to determine the joint angles (thetas) of the manipulator given that the third link should lie on a line an angle Alpha from the x - axis at a known position.

    I believe the solution should be unique but I can't seem to wrap my hands around an equation that proves it.

    To summarize:
    Link lengths: L1, L2, L3
    Angle of line: Alpha
    And it's position (I forgot to label that in the diagram, but say we know the equation of the line y = mx + c)
    Coupling function: Theta3 = Theta2 * k

    Theta1, Theta2, Theta3
    Joint positions, X1,Y1, X2, Y2, X3, Y3

    2. Relevant equations
    Theta3 = Theta2 * k
    Alpha = Theta1 + Theta2 + Theta3 = Theta1 + Theta2(1+k)
    X1 = L1cos(Theta1)
    Y1 = L1sin(Theta1)
    X2 = L1cos(Theta1) + L2cos(Theta1+Theta2)
    Y2 = L1sin(Theta1) + L2sin(Theta1+Theta2)
    X3 = L1cos(Theta1) + L2cos(Theta1+Theta2) + L3cos(Alpha)
    Y3 = L1cos(Theta1) + L2cos(Theta1+Theta2) + L3sin(Alpha)

    3. The attempt at a solution

    I tried assuming I knew one of the joint end positions, i.e X2, Y2, because my intuition tells me there's got to be only one solution. Then maybe I could find another set of equations and somehow cut it out. Here's my work so far:

    By law of cosines:
    X2^2 + Y2^2 = L1^2 + L2^2 - 2L1L2cos(beta)
    where beta is 180 - Theta2 (see diagram)
    beta = acos((L1^2 + L2^2 - (X2^2 + Y2^2)) / (2L1L2))
    Theta2 = 180 - beta
    Theta3 = (180 - beta) * k
    Theta1 = alpha - Theta2 - Theta3

    But now I'm stuck because I can't come up with another good set of equations to remove X2 and Y2 since I don't actually know them. Any insight or advice would be great.


    Attached Files:

    Last edited: Nov 23, 2011
  2. jcsd
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