From 3D velocity to coordinates

I want to calculate the 3D position $T_x, T_y, T_z$ of an object with respect to a coordinate system if I have the mean velocity (norm) $v$ and its 3D rotation $(\omega, \phi, \kappa)$ with respect to the same coordinate system. I want to use a naive motion model, as simple as possible.

So, I assume linear constant velocity and given the time I calculate $T=vt$ where $T$ is the 3D position, $v$ is the mean velocity and $t$ is the time. Then I use sphrerical coordinates to calculate the translation components $T_x=Tsin(\phi)cos(\theta)$
$T_y=Tsin(\phi)sin(\theta)$
$T_z=Tcos(\phi)$

Are the values $\phi$ and $\theta$ the same as the ones from $\omega,\phi,\kappa$? Is it a correct approach, although simple? I don't want to use angular velocity etc.

Any help appreciated!

See Euler Angles.

Give a starting orientation versor $\hat{x}_{initial}$ and the rotation matrix $A$ you can compute the final orientation versor as $\hat{x}_{final} = A \cdot \hat{x}_{initial}$.

$v t$ times this vector will give you the final position, under your hypothesis.

Then you can compute the final position as $\vec{x}_{final} = v t (A \cdot \hat{x}_{initial})$.

Check the link to see how to compute A from $(\omega, \phi, \kappa)$.

• Nice. I usually use the 3D Rotation matrix R, which I guess is the same with A you mentioned. The initial position is (0,0,0) as it is the start of the reference system. Can you provide me a link that explains the last formula x=T(A*x)? Jun 7, 2014 at 17:10
• @Ellie I changed the post a little bit as it was misleading. Anyway, I guess you need to revise your notion of rotation matrix and its usage. It helps you describe the rotation of a body from one orientation to another. I understood your post as: "the final orientation of my body is given by $\hat{x}_{final} = A \cdot \hat{x}_{initial}$. Then let's suppose that it travelled with constant speed $v$ for a time $t$ along that direction. Where will it be now?" Did I understand correctly? If so, everything is stated correctly in my post. Jun 7, 2014 at 17:55
• Yes, you understood correctly. I have the 3D rotation matrix R with respect to the reference point, a mean velocity and time. I need to find the position. Everything is clear at your post, but I have one question. Since my initial position is (0,0,0), everything will be zero at the end? Jun 7, 2014 at 18:02
• Nope: let's say that $v$ is $1 \frac{m}{s}$, $t = 10 \, s$ and that your body initially is oriented towards the x axis: $\hat{x}_{initial} = (1,0,0)$. Let's say that $A$ rotate your body so that it looks towards the z axis $\hat{x}_{final} = A \hat{x}_{inital} = (0,0,1)$. Then you are supposing that it travelled $v t = 10 \, m$ in that direction. Then you'll find it in position $\vec{x}_{final} = 10 (0,0,1) = (0, 0, 10)$ Jun 7, 2014 at 18:26
• Do you mean that $x_{initial}$ includes rotation? From your comment I understood that I shoud do $x_{initial}=(v,0,0)$, right? Jun 7, 2014 at 18:45