Probabilistic Path Planner by Someshwar Marepalli Pratik Desai Ashutosh Sahu Gaurav jain

Motion Planning Definition: Computation of a collision-free path for a movable object among obstacles. Applications: Building Maps Assemble/De-assemble engines Tracking Targets Computational Biology

Challenges in Motion Planning Obstacles Robot geometry Non Holonomic constraints Moving obstacles Uncertain sensing Inexact control

Complete Planners Definition: Planners that solve any solvable problem and return failure for each non solvable one. Real life environments demand Complete Planners. Solving even the basic path planning problem requires time exponential in the robots degrees of freedom. n robots of d degrees of freedom each, complexity becomes exponential in nd. In the presence of moving obstacles, problem becomes exponential in their number. Uncertainties in the robots sensing and control, leads to an exponential dependency on the complexity of the obstacles.

Weaker Complete Planners High complexity of Complete Planners make them impractical for most applications. Heuristic planners have been developed which can solve complicated problems but they fail for some simple ones. Exists a substantial gap between the theoretical worst-case bounds of path planning algorithms and their practical complexity. Design planners that satisfy weaker forms of completeness like Probabilistic and Resolution completeness. Two successful planners which satisfy the former condition are Randomized Path Planner(RPP) and Probabilistic path Planner(PPP).

Randomized Path Planner Potential field U is calculated giving positive and negative values to different configurations. Attracting potential for the goal configuration and repulsive for the obstacles. By descending along U, we attempt to connect the start and the goal configuration. We hit upon a minima at the end of this process and if the minima encountered is the global minima, we have found a solution to the problem, else we perform Brownian motion to achieve it.

Randomized Path Planner (contd) ‏ RPP proves to be very powerful in many cases but it has some drawbacks too, Transition Probabilities P ij ’s are non-trivial in many cases Single shot approach of solving Simple problems consume a lot of time due to low transition probabilities It does not apply directly to non holonomic robots.

Probabilistic Path Planner Very general Planner Builds probabilistic roadmaps by randomly selecting configurations from the free configuration space and then interconnects some pairs by simple feasible paths. Its probabilistically complete and not restricted to any particular class of robots. Consists of two phases, Roadmap construction phase and Query phase.

Roadmap Construction Phase During this phase a data structure is incrementally constructed in a probabilistic way. It is later used in the query phase for solving individual path planning problems Let C f be the free configuration phase Define D C × C R + as the distance measure used Graph G = (V,E) C f × C f 2 is defined by the roadmap construction algorithm.

Roadmap Construction Phase The Roadmap construction algorithm: (1) V = Ø; E = Ø ; (2) loop (3) c = a “randomly” chosen free configuration (4) V = V U {c} (5) N c = a set of neighbours of c chosen from V (6) For all n € N c, in order of increasing D(c,n) do (7) If ¬connected(c,n)^ {L(c,n) Ç C f } then E = E U {(c,n)}

Roadmap Construction Phase Local Planner: It constructs all paths feasible for the robot in the absence of obstacles. Features: It should construct paths in a time efficient manner Probability that local paths obstruct with obstacles should be low. It should guarantee probabilistic completeness of PPP.

Roadmap Construction Phase Neighbors and Edge adding Methods: Number of neighbors (N c ) directly affects the time complexity Attempts to connect nodes that are already in c’s connected component are useless. To avoid too many local planner failures, we only submit pairs of configurations whose relative distance is small (less than a threshold maxdist) ‏ N c C { c` V | D(c,c`) < maxdist }

Roadmap Construction Phase Distance: Should be defined in such a way as to reflect the “failure-chance” well. Rough approximations of sweep volumes are generally preferred. Node Adding Heuristics: It is recommended to guide the node generation by heuristics that create higher node densities in difficult areas.

Query Phase We try to connect the start configuration s and the goal configuration g to s` and g` respectively, subject to the fact that s` and g` belong to the same connected component of G and compute P s`g` Concatenate P ss` P s`g` and P g`g Incase such a path does not exist the query fails (preferably terminate instantaneously)

Query Phase To find the nodes s` and g`, a function query_mapping C C V V is used. Defined as query mapping(a,b) =(a`,b`), such that a` and b` are connected, and D(a,a`) + D(b,b`) = MIN (x,y) W : D(a,x) + D(y, b) ‏ where W = {(x,y) V V |connected(x,y)}

Query Phase Since no obstacle avoidance is incorporated in the Local Planner, it might turn out that the query might fails inspite of a well connected free C-space well. To avoid unnecessarily long paths due to the probabilistic nature, one can smoothen the paths by picking a random path and replacing it by a shorter one.

Thank you !!