I'm studying the kinematic model of a car with a single trailer. This picture shows the system:
I started from the following statements:
With these constraints, I got the Pfaffian form:
And then the kinematic model:
I hope the calculations are good. Now I should use the Lie brackets to verify the system controllability, by checking the accessibility first. To do this, I need to write the system in the following form:
The question is: how do I do it? Is g the two column matrix showed in the model above? In this case, should f be a null vector/matrix? I think this should make sense, but - assuming it's all right - I wouldn't know how to proceed with the Lie brackets, because the iteration I studied leads to the calculation of this sequence: g, [f,g], [f,[f,g]], [f,[f,[f,g]]] and so on. Does it make sense with a null vector as f? In short, the doubt is about how to choose the inputs.
Maybe it's a stupid question, but I didn't really understand the topic and I don't manage to get any concrete results.