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The figure below describes a robot moving in a vertical plane $(x, y)$ with two degrees of freedom: one vertical translation of length $q_1$ and one rotation of angle $q_2$ measured with respect to the horizontal line $x$.

enter image description here

How can we obtain the coordinates $(x_1, y_1)$ and $(x_2, y_2)$ of the points $P_1$ and $P_2$ as functions of $q = (q_1, q_2)$, considering that the point $P_0$ is fixed at the coordinates $(0,0)$ and that the length $P_1 P_2$ is 1 meter?

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Let's call $t$ the angle from the $x$-axis counterclockwise to the arrow from $P_0$ to $P_1$. It's 90 degrees, or $\pi/2$, or whatever you want to call it. But that makes the point $P_1$ be

\begin{align} P_1 &= P_0 + q_1 (\cos t, \sin t) \\ &= (0,0) + q_1 (\cos t, \sin t) \\ &= (0 + q_1 \cos t,0 + q_1 \sin t). \end{align}

Now you can look up the values of sine and cosine to fill in and get the actual answer.

And you can use the same approach to find the coordinates of $P_2$, starting from $P_1$>

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  • $\begingroup$ Thank you for the time! $\endgroup$ – RabbitBadger Jan 21 at 21:25

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