Chapter 4: Chapter 4

Consider a model for the "shopping cart" shown in Figure 4.2 ("knife-edge" or "unicycle" are other names for this example). The state is given by the orientation \(\theta\), together with the coordinates \(x_{1}, x_{2}\) of the midpoint between the back wheels. Figure 4.2: Shopping cart. The front wheel is a castor, free to rotate. There is a non-slipping constraint on movement: the velocity $\left(\dot{x}_{1}, \dot{x}_{2}\right)^{\prime}\( must be parallel to the vector \)(\cos \theta, \sin \theta)^{\prime} .$ This leads to the following equations: $$ \begin{aligned} \dot{x}_{1} &=u_{1} \cos \theta \\ \dot{x}_{2} &=u_{1} \sin \theta \\ \dot{\theta} &=u_{2} \end{aligned} $$ where we may view \(u_{1}\) as a "drive" command and \(u_{2}\) as a steering control (in practice, we implement these controls by means of differential forces on the two back corners of the cart). We view the system as having state space \(\mathbb{R}^{3}\) (a more accurate state space would be the manifold \(\mathbb{R}^{2} \times \mathbb{S}^{1}\) ). (a) Show that the system is completely controllable. (b) Consider these new variables: $z_{1}:=\theta, z_{2}:=x_{1} \cos \theta+x_{2} \sin \theta, z_{3}:=\( \)x_{1} \sin \theta-x_{2} \cos \theta, v_{1}:=u_{2}\(, and \)v_{2}:=u_{1}-u_{2} z_{3}$. (Such a change of variables is called a "feedback transformation".) Write the system in these variables, as \(\dot{z}=\widetilde{f}(z, v) .\) Note that this is one of the systems \(\Sigma_{i}\) in Exercise 4.3.14. Explain why controllability can then be deduced from what you already concluded in that previous exercise.

A distribution on the open subset \(\mathcal{O} \subseteq \mathbb{R}^{n}\) is a map \(\Delta\) which assigns, to each \(x \in \mathcal{O}\), a subspace \(\Delta(x)\) of \(\mathbb{R}^{n}\). A vector field $f \in \mathbb{V}(\mathcal{O})\( is pointwise in \)\Delta\(, denoted \)f \in_{p} \Delta$, if \(f(x) \in \Delta(x)\) for all \(x \in \mathcal{O}\). A distribution is invariant under a vector field \(f \in \mathbb{V}(\mathcal{O})\) if $$ g \in_{p} \Delta \Rightarrow[f, g] \in_{p} \Delta, $$ and it is involutive if it is invariant under all \(f \in_{p} \Delta\), that is, it is pointwise closed under Lie brackets: $$ f \in_{p} \Delta \text { and } g \in_{p} \Delta \quad \Rightarrow \quad[f, g] \in_{p} \Delta . $$ The distribution generated by a set of vector fields $f_{1}, \ldots, f_{r} \in \mathrm{V}(\mathcal{O})\( is defined \)b y$ $$ \Delta_{f_{1}, \ldots, f_{r}}(x):=\operatorname{span}\left\\{f_{1}(x), \ldots, f_{r}(x)\right\\} $$ for each \(x \in \mathcal{O}\). A distribution has constant rank \(r\) if \(\operatorname{dim} \Delta(x)=r\) for all \(x \in \mathcal{O}\).

Suppose that \(\Delta=\Delta_{f_{1}, \ldots, f_{r}}\) is a distribution of constant rank \(r\). Then, 1\. The following two properties are equivalent, for any $f \in \mathbb{V}(\mathcal{O})$ : (a) \(f \in_{p} \Delta\) (b) For each \(x^{0} \in \mathcal{O}\), there are a neighborhood \(\mathcal{O}_{0}\) of \(x^{0}\) and \(r\) smooth functions $\alpha_{i}: \mathcal{O}_{0} \rightarrow \mathbb{R}_{1} i=1, \ldots, r$, so that $$ f(x)=\sum_{i=1}^{r} \alpha_{i}(x) f_{i}(x) \text { for all } x \in \mathcal{O}_{0} $$ 2\. The following two properties are equivalent, for any $f \in \mathbb{V}(\mathcal{O})$ : (a) \(\Delta\) is invariant under \(f\). (b) \(\left[f, f_{j}\right] \in_{p} \Delta\) for each \(j \in\\{1, \ldots, r\\}\). 3\. Finally, the following two properties are equivalent: (a) \(\Delta\) is involutive. (b) \(\left[f_{i}, f_{j}\right] \in_{p} \Delta\) for all $i, j \in\\{1, \ldots, r\\}$.