Equation with two unknowns in complex exponential

[jonasjia]

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.

 

[DrClaude]

One equation and two unknowns. The problem is underdefined and there is more than one solution.

 

[economicsnerd]

If I'm parsing your equation correctly, it can be rearranged to say $$e^{i\theta}= \frac{d}{x+z - ye^{-i\alpha}}$$ which pins down $\theta$ (modulo $2\pi$) for any given value of $\alpha$.

 

[jonasjia]

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.

 

[jonasjia]

to economicsnerd,
thanks for your answer, but it would go further,and get the f(θ) = α