I have some straight and curve pieces that I use to build tracks for a car (robot), so every time the shape of the track where the car will move is known but I don't have its equation (I think getting it will help, but I don't know how).
Every $\delta$ seconds I get the linear velocity of the car $v(t)$ via a sensor (I have no information about the heading and the angular velocity). If we suppose that the car doesn't change the lane, can we compute the position of the car in real time $(x(t),y(t))$ in a Cartesian or any other coordinate system?
I tried Runge–Kutta methods but the results were not that good, I'm looking for a method which can take advantage of the fact that the shape of the track is known.
I thought about using a velocity motion model, but without the angular velocity, it's not going to work.
I was told that I can use a Kalman filter but I don't see how.
Naively you would just integrate the velocity to get the position. The numerical integrators assume you can get the velocity at arbitrarily small time steps, which is not the case. If you are using a velocity sensor it may have drift, where zero velocity is reported as (on average) some small nonzero value. As you integrate over long times this will build up. As you have data at discrete time steps about the best you can do is average the values at neighboring time steps and assume the velocity is constant at that value throughout the time between. You could be a bit more clever because of the known radius of the curves, but I suspect it is not worth while.
As you say, you want to use the knowledge of the shape of the track. I would try to identify the time the car enters and exits each curve. For a first cut, that is when the velocity is nonzero in both axes. You may want to adjust it a bit. If we take the top straight as an example and imagine the car moving from left to right. Find the time the transverse velocity goes to zero and take that as leaving the upper left curve. You know exactly where the car is at that moment. Also find the time the transverse velocity leaves zero, which is when it enters the upper right turn. Again you know where it is. Now integrate the velocity between those points and see if the distance covered matches the known length of the straight. If the computed distance is too short you have to add some time, so move the exit time earlier and the entry time later. You can compute the time change needed by the length error divided by the velocity. Work your way around the track, finding times where you enter and exit each curve.
With this set of known points and times you are set. To get position as a function of time along the top straight I would integrate the velocity from the exit from the top left curve. If it still doesn't come out quite right, add or subtract a bit from all the velocities to make the distance covered along the straight match the known length.