Open problem on risk-aware planning in the plane
We consider the motion-planning problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We recently suggested a natural cost function that balances path length and risk-exposure time. When no risk zones exists, our problem resorts to computing minimal-length paths which is known to be computationally hard in the number of dimensions. It is well known that in two-dimensions computing minimal-length paths can be done efficiently. Thus, a natural question we pose is “Is our problem computationally hard or not?” If the problem is hard, we wish to find an approximation algorithm to compute a near-optimal path. If not, then a polynomial-time algorithm should be found.
We are interested in motion-planning problems where an agent has to compute the least-cost path to navigate through risk zones while avoiding obstacles. Travelling these regions incurs a penalty which is super-linear in the traversal time (see Figure 1. We call the class of problems Risk Aware Motion Planning (RAMP) and use a natural cost function which simultaneously optimizes for paths that are both short and reduce consecutive exposure time in the risk zone.
We are motivated by real-world problems involving risk, where continuous exposure is much worse than intermittent exposure. Examples include pursuit-evasion where sneaking in and out of cover is the preferred strategy, and visibility planning where the agent must ensure that an observer or operator is minimally occluded.
In its general form, our problem can be seen as an instance of the motion-planning problem  which is known to be PSPACE-Hard . Thus, in high-dimensional spaces, a natural approach is to follow the sampling-based paradigm by computing a discrete graph which is then traversed by a path-finding algorithm. Standard path-finding algorithms such as Dijkstra  and A*  cannot be used as optimal plans do not posses optimal substructure. Having said that, we recently suggested efficient path-planning algorithms .
When restricting the planning domain to the two-dimensional plane it is not clear whether the problem is computationally hard or not. It is well known that planning for shortest paths in the plane amid polygonal obstacles can be computed in time, where is the complexity of the obstacles (see  for a survey). When computing shortest paths amid polyhedral obstacles in , or in when there are constraints on the curvature of the path, the problem becomes NP-Hard . Furthermore, the Weighted Region Shortest Path Problem, which is closely related to our problem , is unsolvable in the Algebraic Computation Model over the Rational Numbers . If our problem is computationally hard, as we conjecture, then a reduction, possibly along the lines of  should be provided together with an approximation algorithm. Here, a possible approach would be to sample the boundary of , similar to . For a survey of planning algorithms low dimensions, see, e.g., 
Let be a set of simple pairwise interior-disjoint polygons having n vertices in total. We subdivide into the disjoint sets and which will be used to define the obstacle region and the risk region , respectively. Roughly speaking, these regions are considered to be open sets. However we do not wish to consider points on the boundary of and as points out of the risk region which are collision free. This is captured by the following definition: The forbidden region is the set of all points in the interior of . The risk region is the set of all points in the interior of and all points that lie on the border of and . Finally, the risk-free region is defined as .
A trajectory is a continuous mapping between time and points. We say that is collision free if . The image of a trajectory is called a path. Given a trajectory , and some time , let be the latest time such that . Notice that if then . We define the current exposure time of at as . Namely, if then is the time passed since last entered . If then .
We are now ready to define our cost function. Let be a trajectory and any function such that and . The cost of , denoted by is defined as
Equation 1 penalizes continuous exposure to risk in a super-linear fashion (hence the requirement that ). As , the cost of traversing the risk-free region is simply path length. See Figure 2 for a conceptual visualization of the current exposure time and our cost function.
Equipped with our cost function we can formally state the risk-aware motion-planning problem:
Planar Risk-aware motion-planning problem (pRAMP) Given the tuple , where are start and goal points, compute with the set of all collision-free trajectories connecting and
We defined our problem to be as general as possible. However, to simplify the discussion, we assume that the robot is moving in constant speed and we use . Thus, we can re-write Equation 1 as
Using the assumption that the robot is moving in constant speed, we can use the terms duration of a trajectory and path length interchangeably (here we measure path length as the Euclidean distance). Further exploiting this assumption and by a slight abuse of notation we can also use Equation 2 to define the cost of a path (and not of a trajectory). For different properties of this cost function, the reader is referred to .
3Discussion and open questions
When considering the complexity of a planning problem, one needs to consider both the algebraic complexity and the combinatorial complexity. If we use the Algebraic Computation Model over the Rational Numbers (ACM), then we conjecture that the problem is unsolvable. A proof may follow the lines taken in  for the Weighted Region Shortest Path Problem.
Assessing the combinatorial complexity of our problem, defined analogously to the number of “edge sequences” is not as straightforward. Several hardness results for planning problems use reductions from 4CNF-satisfiability . The proofs use the idea of “path encoding” which involves constructing an environment that admits an exponential number of distinct shortest paths between and . Each path is associated with a truth assignment of a given formula . Then, the environment is augmented with additional obstacles that block every path whose associated truth assignment does not satisfy the formula . The underlying problem with using this approach is that in the plane it depends heavily on the fact that a minimal-cost paths can self-intersect, which is not the case in our setting.
Assuming that the problem is computationally hard, we seek an approximation algorithm such that given some returns a path whose cost is at most the cost of the optimal path in time polynomial in and . A natural approach would be to sample densely along the boundary of and compute the visibility graph defined over the sampled points and the vertices in . A minimal-cost path may then be computed in polynomial time [?]. However, the running time of this algorithm also depends on the length of the edges of polygons in (see similar approach and analysis in ).
We believe that a possible approach would be to sample the boundary of more carefully, similar to .
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