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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:
Knowns:
Link lengths: L1, L2, L3
Angle of line: Alpha = Theta1 + Theta2 + Theta3 = Theta1 + Theta2(1+k)
And it's position
Coupling function: Theta3 = Theta2 * k
Unknowns
Theta1, Theta2, Theta3
Joint positions, X1,Y1, X2, Y2, X3, Y3
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.
Thanks!
I believe the solution should be unique but I can't seem to wrap my hands around an equation that proves it.
To summarize:
Knowns:
Link lengths: L1, L2, L3
Angle of line: Alpha = Theta1 + Theta2 + Theta3 = Theta1 + Theta2(1+k)
And it's position
Coupling function: Theta3 = Theta2 * k
Unknowns
Theta1, Theta2, Theta3
Joint positions, X1,Y1, X2, Y2, X3, Y3
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.
Thanks!
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