## 7.1.1 Problem Formulation

The formulation is designed to allow the tools and concepts learned so far to be applied directly. Let denote the time interval, which may be bounded or unbounded. If is bounded, then , in which 0 is the initial time and is the final time. If is unbounded, then . An initial time other than 0 could alternatively be defined without difficulty, but this will not be done here.

Let the state space be defined as , in which is the usual C-space of the robot, as defined in Chapter 4. A state is represented as , to indicate the configuration and time components of the state vector. The planning will occur directly in , and in many ways it can be treated as any C-space seen to far, but there is one critical difference: Time marches forward. Imagine a path that travels through . If it first reaches a state , and then later some state , some traveling backward through time is required! There is no mathematical problem with allowing such time travel, but it is not realistic for most applications. Therefore, paths in are forced to follow a constraint that they must move forward in time.

Now consider making the following time-varying versions of the items used in Formulation 4.1 for motion planning.

Formulation 7..1 (The Time-Varying Motion Planning Problem)
1. A world in which either or . This is the same as in Formulation 4.1.
2. A time interval that is either bounded to yield for some final time, , or unbounded to yield .
3. A semi-algebraic, time-varying obstacle region for every . It is assumed that the obstacle region is a finite collection of rigid bodies that undergoes continuous, time-dependent rigid-body transformations.
4. The robot (or , , for a linkage) and configuration space definitions are the same as in Formulation 4.1.
5. The state space is the Cartesian product and a state is denoted as to denote the configuration and time components. See Figure 7.1. The obstacle region, , in the state space is defined as (7.1)

and . For a given , slices of and are obtained. These are denoted as and , respectively, in which (assuming is one body) (7.2)

and .

6. A state is designated as the initial state, with the constraint that for some . In other words, at the initial time the robot cannot be in collision.
7. A subset is designated as the goal region. A typical definition is to pick some and let , which means that the goal is stationary for all time.
8. A complete algorithm must compute a continuous, time-monotonic path, , such that and , or correctly report that such a path does not exist. To be time-monotonic implies that for any such that , we have , in which and . Example 7..1 (Piecewise-Linear Obstacle Motion)   Figure 7.1 shows an example of a convex, polygonal robot that translates in . There is a single, convex, polygonal obstacle . The two of these together yield a convex, polygonal C-space obstacle, , which is shown for times , , and . The obstacle moves with a piecewise-linear motion model, which means that transformations applied to are a piecewise-linear function of time. For example, let be a fixed point on the obstacle. To be a linear motion model, this point must transform as for some constants . To be piecewise-linear, it may change to a different linear motion at a finite number of critical times. Between these critical times, the motion must remain linear. There are two critical times in the example. If is polygonal, and a piecewise-linear motion model is used, then is polyhedral, as depicted in Figure 7.1. A stationary goal is also shown, which appears as a line that is parallel to the -axis. In the general formulation, there are no additional constraints on the path, , which means that the robot motion model allows infinite acceleration and unbounded speed. The robot velocity may change instantaneously, but the path through must always be continuous. These issues did not arise in Chapter 4 because there was no need to mention time. Now it becomes necessary.7.1

Steven M LaValle 2020-08-14