Boundary sensors

If the state space has an interesting boundary, as in the case of ${\cal C}_{free}$ for motion planning problems, then many important boundary sensors can be formulated based on the detection of the boundary. Figure 11.12 shows several interesting cases on which sensors are based.

Suppose that the state space is a closed set with some well-defined boundary. To provide a connection to motion planning, assume that $X = \operatorname{cl}({\cal C}_{free})$, the closure of ${\cal C}_{free}$. A contact sensor determines whether the boundary is being contacted. In this case, $Y = \{0,1\}$ and $h$ is defined as $h(x) = 1$ if $x \in \partial X$, and $h(x) = 0$ otherwise. These two cases are shown in Figures 11.12a and 11.12b, respectively. Using this sensor, there is no information regarding where along the boundary the contact may be occurring. In mobile robotics, it may be disastrous if the robot is in contact with obstacles. Instead, a proximity sensor is often used, which yields $h(x) = 1$ if the state or position is within some specified constant, $r$, of $\partial X$, and $h(x) = 0$ otherwise. This is shown in Figure 11.12.

In robot manipulation, haptic interfaces, and other applications in which physical interaction occurs between machines and the environment, a force sensor may be used. In addition to simply indicating contact, a force sensor can indicate the magnitude and direction of the force. The robot model must be formulated so that it is possible to derive the anticipated force value from a given state.