First off, this is not a homework question. I am simply trying to figure out how to make a discrete time model into a continuous one. I don't expect anyone to work this out for me, I just want to start heading in the right direction.(adsbygoogle = window.adsbygoogle || []).push({});

I have a model of a bicycle on a 2d plane that has a position (x,y) and an orientation (theta). Theta is with respect to the X axis. The bike steers with the front wheel at an angle (alpha) with respect to the orientation (theta). The distance between the rear wheel and the front wheel is L. Now, given an initial state (x,y,theta), I am given a distance the bike will travel (d) at a certain steering angle (alpha). It can be shown that the new position and orientation of the bike is as follows:

x' = cx + sin(theta+Beta)*R

y' = cy - cos(theta+Beta)*R

theta' = (theta+Beta) (%mod 2pi optionally)

where

cx = x - sin(theta)*R

cy = y + cos(theta)*R

R = d/Beta

Beta = d*tan(alpha)/L

Now what I want to do is generate a trajectory of the bicycle given an arbitrary time varying function for both the steering angle (alpha) and the instantaenous velocity d(d)/dt. How would I go about starting a problem like this?

If we ignore all the details that I listed above, I should be able to write the new position and orentation as a vector:

| x' |

| y' | = f(x,y,theta,alpha,d)

|theta|

What steps do I need to take next? When I move to a continuous model, all the variables become time varying: f(x(t),y(t),theta(t),d(t)). My first thought is that I take the derivative of each component with respect to alpha and d. I would then end up integrating from time zero to time T given the initial state vector and given time varying functions for alpha and d.

Thanks

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# Modeling a bicycle

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