14.1.3.1 Additional constraints on phase variables

In many applications, additional constraints may exist on the phase variables. These are called phase constraints and are generally of the form $h_i(x) \leq 0$. For example, a car or hovercraft may have a maximum speed for safety reasons. Therefore, simple bounds on the velocity variables will exist. For example, it might be specified that $\Vert{\dot q}\Vert \leq {\dot q}_{max}$ for some constant ${\dot q}_{max} \in (0,\infty)$. Such simple bounds are often incorporated directly into the definition of $X$ by placing limits on the velocity variables.

Figure 14.2: In the NASA/Lockheed Martin X-33 re-entry problem, there are complicated constraints on the phase variables, which avoid states that cause the craft to overheat or vibrate uncontrollably. (Courtesy of NASA)

In other cases, however, constraints on velocity may be quite complicated. For example, the problem of computing the re-entry trajectory of the NASA/Lockheed Martin X-33 reusable spacecraft^14.2 (see Figure 14.2) requires remaining within a complicated, narrow region in the phase space. Even though there are no hard obstacles in the traditional sense, many bad things can happen by entering the wrong part of the phase space. For example, the craft may overheat or vibrate uncontrollably [160,201,662]. For a simpler example, imagine constraints on $X$ to ensure that an SUV or a double-decker tour bus (as often seen in London, for example) will not tumble sideways while turning.

The additional constraints can be expressed implicitly as $h_i(x) \leq 0$. As part of determining whether some state $x$ lies in ${X_{free}}$ or ${X_{obs}}$, it must be substituted into each constraint to determine whether it is satisfied. If a state lies in ${X_{free}}$, it will generally be called violation-free, which implies that it is both collision-free and does not violate any additional phase constraints.