Open problem on risk-aware planning in the plane

Open problem on risk-aware planning in the plane

Abstract

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.

1Introduction

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.

Figure 1:   Risk aware motion planning. We need to plan a minimal-cost connecting s and g while avoiding obstacles (red). Our cost function penalizes continous exposure to the risk regions (purple), thus the optimal path (solid green) favours intermittent exposure over the long exposure taken by the shortest path (dotted blue).
Figure 1: Risk aware motion planning. We need to plan a minimal-cost connecting and while avoiding obstacles (red). Our cost function penalizes continous exposure to the risk regions (purple), thus the optimal path (solid green) favours intermittent exposure over the long exposure taken by the shortest path (dotted blue).

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 [10] which is known to be PSPACE-Hard [13]. 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 [6] and A* [8] cannot be used as optimal plans do not posses optimal substructure. Having said that, we recently suggested efficient path-planning algorithms [14].

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 [12] 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 [3]. Furthermore, the Weighted Region Shortest Path Problem, which is closely related to our problem [11], is unsolvable in the Algebraic Computation Model over the Rational Numbers [5]. If our problem is computationally hard, as we conjecture, then a reduction, possibly along the lines of [3] should be provided together with an approximation algorithm. Here, a possible approach would be to sample the boundary of , similar to [1]. For a survey of planning algorithms low dimensions, see, e.g., [7]

2Problem formulation

Figure 2:   Relation between a trajectory \gamma(t) (top), recent exposure time \lambda_\gamma(t) (middle) and cost c_f(\gamma(t)) (bottom) as a function of time. In t \in [0,t_1], \gamma stays in \ensuremath{\mathcal{X}}\xspace_{\rm safe}, hence \lambda_\gamma(t)=0 and the cost grows linearly with time. At t=t_1, \gamma enters \ensuremath{\mathcal{X}}\xspace_{\rm risk}, \lambda_\gamma(t) grows linearly and the cost grows super-linearly. At t=t_2, \gamma leaves \ensuremath{\mathcal{X}}\xspace_{\rm risk}, \lambda_\gamma(t)=0 and the cost returns to growing linearly.
Figure 2: Relation between a trajectory (top), recent exposure time (middle) and cost (bottom) as a function of time. In , stays in , hence and the cost grows linearly with time. At , enters , grows linearly and the cost grows super-linearly. At , leaves , and the cost returns to growing linearly.

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 [14].

3Discussion and open questions

3.1Hardness

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 [5] 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 [2]. 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.

3.2Approximation algorithm

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 [15]).

We believe that a possible approach would be to sample the boundary of more carefully, similar to [1].

References

  1. 2016.
    Agarwal, P. K.; Fox, K.; and Salzman, O. An efficient algorithm for computing high-quality paths amid polygonal obstacles.
  2. 2003.
    Asano, T.; Kirkpatrick, D. G.; and Yap, C. Minimizing the trace length of a rod endpoint in the presence of polygonal obstacles is np-hard.
  3. 1987.
    Canny, J. F., and Reif, J. H. New lower bound techniques for robot motion planning problems.
  4. 2005.
    Choset, H.; Lynch, K. M.; Hutchinson, S.; Kantor, G.; Burgard, W.; Kavraki, L. E.; and Thrun, S. Principles of Robot Motion: Theory, Algorithms, and Implementation.
  5. 2014.
    De Carufel, J.-L.; Grimm, C.; Maheshwari, A.; Owen, M.; and Smid, M. A note on the unsolvability of the weighted region shortest path problem.
  6. 1959.
    Dijkstra, E. W. A note on two problems in connexion with graphs.
  7. 2016.
    Halperin, D.; Salzman, O.; and Sharir, M. Algorithmic motion planning.
  8. 1968.
    Hart, P. E.; Nilsson, N. J.; and Raphael, B. A formal basis for the heuristic determination of minimum cost paths.
  9. 2011.
    Kirkpatrick, D. G.; Kostitsyna, I.; and Polishchuk, V. Hardness results for two-dimensional curvature-constrained motion planning.
  10. 2006.
    LaValle, S. M. Planning algorithms.
  11. 1991.
    Mitchell, J. S. B., and Papadimitriou, C. H. The weighted region problem: Finding shortest paths through a weighted planar subdivision.
  12. 2016.
    Mitchell, J. S. B. Shortest paths and networks.
  13. 1979.
    Reif, J. H. Complexity of the mover’s problem and generalizations (extended abstract).
  14. 2017.
    Salzman, O.; Hou, B.; and Srinivasa, S. S. Efficient motion planning for problems lacking optimal substructure.
  15. 2008.
    Wein, R.; van den Berg, J. P.; and Halperin, D. Planning high-quality paths and corridors amidst obstacles.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
31161
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description