difference between non-holonomic and linear control systems

Question

I already asked this question in the mathoverflow forum, but no one could answer. Probably it is not a question of mathematics but robotics.

I'm struggling with the definitions and differences between a non-holonomic and a linear control systems.

I'm working with a phase space $P\subset R^{m}$ and a control space $U\subset R^{l}$ and my control system

$\dot{p}=f(p,u)$

is of the form

$\dot{p}=\sum^{l}_{i=1}u_{i}P_{i}(p)$

where $P_{1},\dots,P_{l}$ are smooth vector fields on $P$.

In Frederic Jeans book Definition 1.1 a control system of this form is called non-holonomic. I already read that there is the adjective "linear" which would fit better in my opinion since whenever I use the form of the control system I use the fact that it depends linearly on the control.

Can someone advice me literature where it is stated like that?

When I read more about non-holonomicity I found that it is a question whether the vectorfields $P_{i}$ span the whole $TP$.

Can someone explain how linearity and non-holonomicity are connected?

Answer 1

I have to admit, I have no experience with non-holomonic CONTROL System. I have dealt with non-holomonic MECHANICAL systems.

As defined here, a non-holominc mechanical system annot move in arbitrary directions inits configuration space. A mundane example is a car. It can occupy and X, Y position and Z rotation on a flat surface, but it cannot move laterally in the Y direction. It has to manuver to get there.

In mobile robotics, simplisticly this problem is dealt with at a higher level then the closed loop control system. It is considered a constraint for the path/trajectory planner (open loop). After the path/trajectory has been planned, the setpoints are given to the closed loop control system (cyclically), which can be a linear control system.

I understand that there is research in Optimal Control Theory, which includes the non holomonic planning in the closed loop as an optimization problem, e.g. this paper and this paper, but i cannot offer any insight on the intricacies of these methods.