I have two robotic links. They're basically sticks. These two linked sticks can be controlled by changing their angle with respect to the former link.
In this case, the first stick simply moves like a clock's hand. The second one also moves like a clock's hand, but with the endpoint of the first hand as a pivot.
What I'm trying to get is a way to describe movement is such fashion that the speed in the vertical dimension is zero. Up to now I've described the endpoint as
$$ p = \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \cos(\alpha) + \cos(\beta-\alpha) \\ \sin(\alpha) + \sin(\beta-\alpha) \end{bmatrix} $$
and to make the vertical movement zero, the derivative of the vertical component needs to be naught, so
$$ \frac{d}{dt}(\sin(\alpha) + \sin(\beta-\alpha)) = \dot\alpha \cos(\alpha) + (\dot\beta-\dot\alpha)\cos(\beta-\alpha) = 0 $$
with horizontal movement as
$$ \frac{d}{dt}(\cos(\alpha) + \cos(\beta-\alpha)) = \dot\alpha \sin(\alpha) + (\dot\beta-\dot\alpha)\sin(\beta-\alpha) = m(t) $$
where $m(t)$ is just a function describing the speed in $x$. It may simply be a constant.
However, I'm stuck here. I tried putting the equations into Wolfram, but it didn't give me a result. As a hint of something I noticed is that whenever $\alpha(t) = ct,\ \beta(t) = 2ct$ the endpoint moves radially in- or outwards.
Thanks for your help!