# I Velocities when generating an Archimedean spiral trajectory

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1. Aug 6, 2017

### Benyoucef Rayane

hey, I just want to know, if I am to send a velocity commands to generate a spiral trajectory, What would be these velocities (angular and linear)??

2. Aug 6, 2017

3. Aug 6, 2017

4. Aug 6, 2017

### rumborak

Since the usual definition of the Archimedes Spiral takes the angle as the input, it means you choose the angular velocity ω. Regarding linear velocity, I don't have an exact formula at hand, but since it gets closer to a circle with increasing angle, the linear velocity will asymptomatically approach ωr.

5. Aug 6, 2017

For the OP @Benyoucef Rayane A google shows the Archimedes spiral has $r=\theta^a$ with $a=1$. This means $r=\theta$ for this spiral. We can write the velocity $\vec{v}=(\frac{dr}{dt}) \hat{a}_r+(r \dot{\theta}) \hat{a}_{\theta}$. We have for $r=\theta$, that $\frac{dr}{dt}=\dot{\theta}=\omega$. This gives $\vec{v}=\omega \hat{a}_r+(r \omega) \hat{a}_{\theta}$. As $r$ gets large, $\vec{v} \approx (r \omega ) \hat{a}_{\theta}$ as @rumborak pointed out.

6. Aug 12, 2017

### Benyoucef Rayane

Thanks guys, I got it now.