How do I compute the translation and rotation velocities of a robot

Question

I am stuck at computing the translation and rotation of a robot moving onto an ellipse given by:

$$ p(t) =(m) + \cos(t)∗a + \sin(t)*b $$

where m = center of ellipse and the two axes a (horizontal) and b (vertical).

At the moment I am using derivation $p'(t)$ for calculating the translation, like

$$ p'(t)_x = -\sin(t)*a \\ p'(t)_y = \cos(t)*b $$

$$ p'(t) = \sqrt{p'(t)_x^2 + p'(t)_y^2} $$

and rotation velocity, with

$$ p''(t)_x = -\cos(t)*a \\ p''(t)_y = -\sin(t)*b $$

$$ p''(t) = \frac{p'(t)_x p''(t)_y - p''(t)_x p'(t)_y}{ (p'(t)_x)^2 + (p'(t)_y)^2} $$

(see formula)

However, the robot is just moving in strange curls instead of an ellipse. If anyone needs the full code just say so.

$$ \omega = \dot{\phi} = \frac{\dot{d_x}\ddot{d_y} - \ddot{d_x}\dot{d_y}}{\dot{d_x}^2+\dot{d_y}^2} $$

Answer 1

As Chuck said you are using the parametric form of the ellipse

$p(t) = (m) + cos(t)*a + sin(t)*b$ the $t$ doesn't represent the time but the eccentric anomaly. But from this equation you can create a matrix for $t\epsilon [0,2\pi]$ and calculate the points on the eclipse from the equations

$ x = a*cos(t) $ , $ y = b*sin(t) $ then you have $(x,y)$ points and by setting a specific time that you want to go from one point to another you can find the velocities ${\Delta x\over \Delta t} , {\Delta x\over \Delta t} $. Not the best way to do it, but it is something.