14.6.3.2 Determining the allowable accelerations

The actions in $U$ that cause $\tau$ to be followed can now be characterized. An action $u \in U$ follows $\tau$ if and only if every equation of the form (14.39) is satisfied. If $d\tau_i/ds \not = 0$ for all $i$ from $1$ to $n$, then $n$ such equations exist. Suppose that $u_1$ is chosen, and the first equation is solved for ${\ddot s}$. The required values of the remaining action variables $u_2$, $\ldots$, $u_n$ can be obtained by substituting the determined ${\ddot s}$ value into the remaining $n-1$ equations. This means that the actions that follow $\tau$ are at most a one-dimensional subset of $U$.

If $d\tau_i/ds = 0$ for some $i$, then following the path requires that ${\dot q}_i = 0$. Instead of (14.39), the constraint is that $h_i(q,{\dot q},u) = 0$. Example 14.6 will provide a simple illustration of this. If $d\tau_i/ds = 0$ for all $i$, then the configuration is not allowed to change. This occurs in the degenerate (and useless) case in which $\tau$ is a constant function.

In many cases, a value of $u$ does not exist that satisfies all of the constraint equations. This means that the path cannot be followed at that particular state. Such states should be removed, if possible, by defining phase constraints on $X$. By a poor choice of path $\tau$ violating such a phase constraint may be unavoidable. There may exist some $s$ for which no $u \in U$ can follow $\tau$, regardless of ${\dot s}$.

Figure 14.26: A bad path for path-constrained trajectory planning.

Even if a state trajectory may be optimal in some sense, its quality ultimately depends on the given path $\tau: [0,1] \rightarrow {\cal C}_{free}$. Consider the path shown in Figure 14.26. At $\tau(1/3)$, a ``corner'' is reached. This violates the differentiability assumption and would require infinite acceleration to traverse while remaining on $\tau$. For some models, it may be possible to stop at $\tau(1/3)$ and then start again. For example, imagine a floating particle in the plane. It can be decelerated to rest exactly at $\tau(1/3)$ and then started in a new direction to exactly follow the curve. This assumes that the particle is fully actuated. If there are nonholonomic constraints on ${\cal C}$, as in the case of the Dubins car, then the given path must at least satisfy them before accelerations can be considered. The solution in this case depends on the existence of decoupling vector fields [157,224].

It is generally preferable to round off any corners that might have been produced by a motion planning algorithm in constructing $\tau$. This helps, but it still does not completely resolve the issue. The portion of the path around $\tau(2/3)$ is not desirable because of high curvature. At a fixed speed, larger accelerations are generally needed to follow sharp turns. The speed may have to be decreased simply because $\tau$ carelessly requires sharp turns in ${\cal C}$. Imagine developing an autonomous double-decker tour bus. It is clear that following the curve around $\tau(2/3)$ may cause the bus to topple at high speeds. The bus will have to slow down because it is a slave to the particular choice of $\tau$.