# Serial Link chain with constrained geometry

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:
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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Baluncore
2021 Award
I believe the solution should be unique but I can't seem to wrap my hands around an equation that proves it.
The link lengths are known and fixed. L1, L2 and L3.
Given the specified point x3, y3 and alpha, we can compute the position of x2, y2.

There are then two circles, one has radius L1, with centre at the origin.
The other circle has radius L2, with centre at x2, y2.
The intersection points of the two circles, (if they do intersect), give the two possible solutions for x1, y1.

Unfortunately the theta3 = k * theta2 constraint has not been satisfied.
It is clear that movement of link3 along the alpha line will force the value of k to be changed.
So the system is over-constrained.
Which explains why the equations cannot be solved.