# Controllability of a car with a single trailer

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.

• If i have understood the question right, the goal is to make a system identification for the Truck backer upper problem. With mathematical equations, the forward model is generated which contains of three different points in the 2d-space: p1=trailer, p2=car, p3=carwheel. If the player is sending an action to the model, it will produce the new position of the points. And the question was, if the error of the forward model is low, so it will be equal to the reality. – Manuel Rodriguez Mar 24 '19 at 18:44
• It seems very similar to this system. I didn't know that kind of problem, thank you, I will look for some scientific articles about it. – Silmaar Mar 24 '19 at 21:50
• If I understand correctly, you have a complete model in which you have listed all your veriables. Isn't the $\dot{q} = f(x) + g(x)u$ about deviding into control signal related part and control signal independent part? – 50k4 Mar 26 '19 at 9:56
• Exactly. Let's suppose I want u to be the steering angle, which is (thetaA-thetaP). I think it's the most reasonable choice. I still don't know how to divide one part from the other one, because in that case I'd have sin(u) and cos(u). – Silmaar Mar 26 '19 at 10:34

"A car with n trailers" is known to be a differentially flat system. Flatness implies that the Lie algebra generated by the system's vector fields (f and g_i's) is full. Therefore, the example you give is controllable.

In the example, f is zero, but you have two vector fields g1 and g2 that are associated with u1, u2, respectively. You need to work on the Lie brackets that are generated by g1 and g2.

x = g1*u1 + g2*u2;

I'd like to point out some references on the subject. You can find more information how Lie algebra is used for your system.

[1] Rouchon, P., Fliess, M., Lévine, J., & Martin, P. (1993, June). Flatness and motion planning: the car with n trailers. In Proc. ECC’93, Groningen (pp. 1518-1522).

[2] M. Fliess, J. L´evine, Ph. Martin, and P. Rouchon. D´efaut d’un systeme non lin´eaire et commande haute fr´equence. C.R. Acad. Sci. Paris, I-316, 1993

[3] J.P. Laumond. Controllability of a multibody mobile robot. In IEEE International Conf. on advanced robotics, 91 ICAR, pages 1033–1038, 1991.

Robot control problems are usually treated as computational problems. The software engineers are trying to realize an algorithm which is searching in the state space. So called pathplanning and motion planning algorithms are used to bring a robot-system into a desired state. The disadvantage of this classical approach is, that no mathematics but programming skills are important. It ignores the mathematical basics. The better way is to use a modern university driven mathematics to describe robot control problems in an elegant way. No digital computer is needed here.

The question implies that the robot control problem can be described with Lie algebra. Lie Algebra is an example for a topological group for transforming a nonlinear problem into a linear one. The first subquestion which has to be answered is, if Lie groups are powerful enough, which means if they are Turing-complete? The answer is yes, if we are talking about quantum Turing machines.

Now we can go on to solve the question itself. For controlling the robot trailer problem we have to create a recursive algorithm which is executed on a hypothetical quantum computer. The algorithm is using lie groups to determine the Lagrangian equations of motion.

Literature If the idea is to teach lie brackets without mention neural turing machine, ignoring quantum computers and stack-based computation. The resulting request for getting the right literature in Google Scholar is

"lie group" -"quantum computer" -"turing machine"
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The minus sign allows to exclude a keyword from the result list.

• I should use Lie Algebra to deal with the problem because I studied that kind of solution, and yes, it should be powerful enough. Anyway, I'm solving Lie brackets with a proper software and I know how to do those calculations, I tried with random data and it works. I do not know how to get to the right form of the system, instead (i.e. functions f and g above). – Silmaar Mar 26 '19 at 9:05
• Reducing Lie algebra to lie brackets isn't the way it works. The question is not how to transform equation 1 into another one with matlab, but the question has to do with implementing lie groups on a differential turing machine. If i interpret the literature correct, which was published in the last two years at the arxiv archive, then it's an advanced topic not researched before. I would guess, that the nobel price or something higher would be awarded if somebody is able to answer the question right. – Manuel Rodriguez Mar 26 '19 at 9:22
• Maybe I just explained myself in the wrong way, I hope the question isn't so difficult. I don't even know how a differential Turing machine works, this should be just an exercise with some calculations to do. It asks if the system is controllable and observable, I don't need the Lagrangian equations of motion. But I can't find help in the literature, so maybe you understood my problem and it's as you say. I really don't know. – Silmaar Mar 26 '19 at 9:36