Planning High-Quality Paths and Corridors Ron Wein Jur P. van den Berg (U. Utrecht) Dan Halperin

Part I: The Visibility- Voronoi Complex and Its Applications

Motivation Plan a “natural looking” collision-free motion path for a robot translating on the plane amidst a set P 1, …, P m of configuration- space obstacles, given start and goal configurations s and g. By “natural looking” we mean the path should be: Short – not containing unnecessary long detours. Having some clearance – not getting to close to an obstacle. Smooth – not containing sharp turns ( C 1 -smooth).

Application Motion planning for coherent groups of entities (computing a backbone path). Kamphuis and Overmars (2004)

Visibility Graphs Used to plan shortest paths. Constructed in O(n 2 log n) time, where n is the total number of vertices. Output-sensitive O(n log n + S) algorithms also exist. Query time is O(n log n + S), using Dijkstra’s algorithm. Resulting paths have no clearance Lozano-Pérez (1979)

The “Retraction” Method Construct the Voronoi diagram of the obstacles in O(n log n). Query time is O(log n + k). Paths may be too long and may contain sharp turns Ó’Dúnglaing and Yap (1985), Rohnert (1991)

The VV (c) -Diagram 1. Dilate the obstacles (Minkowski sums). 2. Compute visibility edges between dilated vertices. 3. Add Voronoi chains between chain-points. 4. Compute visibility edges emanating from chain points.

Properties of the VV (c) -Diagram Outputs shortest paths which keep the preferred amount of clearance from the obstacles, where possible. Paths are smooth. In case of narrow passages (between chain points), we get a path with the maximal possible clearance (locally).

The Evolution of the VV (c) -Diagram c = 0, …, ½.

The VV-Complex The VV (c) -diagram interpolates between the visibility graph and the Voronoi diagram of the obstacles. The VV-complex encodes all VV (c) -diagrams for all c values. Easily queried for any given c, without the need to construct the VV (c) -diagram. Constructed in O(n 2 log n) time, handling  (n 2 ) events. c = 0c > 0 c = 

The VV-Complex (II) Suppose the clearance value grows from 0 to . We can associate with each edge e we encounter a validity range R(e) = [c min (e), c max (e)] of c-values for which this edge is valid. The VV-complex of the scene of polygonal obstacles P 1, …, P m is therefore: The Voronoi diagram V of the obstacles. A set T of interval trees: For each obstacle vertex u, T (u) stores the incident edges to u indexed by their validity range Similarly, there is also a tree T (p) for each chain point p.

Bitangents to Dilated Vertices u v uv rr uv ll uv rl uv lr For each obstacle vertex u we keep two circular lists L l (u) and L r (u) of “left” and “right” bitangents, sorted by their slopes.

Computing the VV-Complex (Initialization) 1.Compute the visibility graph of the obstacles P 1, …, P m. 2.Examine each bitangent edge in the visibility graph: Assign 0 to be the minimum value of its validity range. 3.Initialize an empty event queue Q. 4. Construct L l (u) and L r (u) for each obstacle vertex u. 5.Compute the initial visibility events based on adjacencies in these lists. 6.Compute the Voronoi diagram of P 1, …, P m. 7.For each Voronoi arc a that contains the minimum clearance value c min of its chain  a, insert the chain event  c min, a  into Q.

Visibility Events Occur when two edges uv and uw become equally sloped. We assign a maximal value for the validity range of the blocked edges. Some edges may become valid, and we assign a minimal value for their validity ranges. There are  (n 2 ) visibility events, each takes O(log n) to handle. u v w uw rl uv rl u v w uv ll uw lr

Chain Events Occur when a chain start appearing in the VV (c) -diagram. We create two chain points associated with this chain. There are  (n) chain events, each takes O(n log n) to handle. The motion of the chain points along the chain causes tangency events and endpoint events. All these events are handled at O(n 2 log n) time in total. aa aa aa p2(a)p2(a) p1(a)p1(a)

“Life-Cycle” of an Edge

Querying the VV-Complex 1.For each Voronoi chain compute the chain points (two at most) that correspond to the given c-value. 2.Perform radial sweep from s and from g and obtain their incident visibility edges. 3.Execute Dijkstra’s algorithm from s. The graph is implicitly maintained, as we obtain the incident edges of each vertex x we encounter from T (x). We do this until reaching g. The total query time is O(n log n + k), where k is the number of edges encountered during the search. Given a start configuration s, a goal configuration g and a preferred clearance value c’:

Implementation and Experiments We have implemented a C GAL -based application that can robustly construct the VV (c) -diagram for a given c-value. We employ: C GAL ’s segment Voronoi diagram package (Karavelas). C GAL ’s arrangement package with the conic-arc traits (W.). The G MP and C ORE number types. Diagram construction takes 3–60 seconds. Query time is 0.02—0.1 seconds (compared with 0.5–1 seconds that a smoothing phase would consume).

Part II: Planning Near- Optimal Corridors amidst Obstacles

The Concept of Corridors Instead of making the moving object(s) follow a 1-dimensional motion path, we let them move in a corridor indicating the general direction for the motion. Inside the corridor, motion is determined by a local planner (e.g. using the potential field method). s g

Applications For many applications it is better to use a corridor, which allows more flexibility in comparison to having a 1-dimensional path: coherent group motion, small moving obstacles (and large static ones), planning camera motions following a moving character, avoiding predictable motions (in games), allowing non-holonomic and kineodynamic planning.

A corridor C =  (t), w(t), w max  in a d-dimensional workspace is the union of the balls: Where:  : [0, L]   d is the backbone path, w: [0, L]  (0, w max ] defines the corridor width, w max is the preferred width. Defining a Corridor

Given a corridor C =  (t), w(t), w max , its weighted length, denoted L * (C), is given by: The Weighted-Length Measure We can improve the quality of a corridor by either making it wider (up to the preferred width), or by making its backbone path shorter.

Observation: The width function of an optimal corridor is given by: where c(p) is the clearance of the point p (its distance from the nearest obstacle). Properties of an Optimal Corridor Lemma: The backbone path of an optimal corridor connecting any given start and goal positions s and g is smooth. p

Refraction of an Optimal Path Snell’s Law: If the width function is not continuous, then the backbone path of the optimal corridor connecting p 1 and p 2 satisfies: p1p1 p2p2 11 22 w1w1 w2w2

Optimal Corridors amidst Point Obstacles We consider planar environments cluttered with point obstacles p 1, …, p n   2 and a preferred width w max. Let us first assume that the points are well-separated with respect to w max, that is the discs B(p i ; w max ) are pairwise disjoint in their interiors (namely ||p i – p j ||  2w max ). We first examine what happens when the two endpoints are both in the vicinity of a single obstacle p (inside B(p; w max )).

p r1r1 r2r2 11 22 A Using Snell’s Law: Near a Single Point Obstacle 11 B The Law of Sines on  pAB: Hence:  1 =  1.

The Logarithmic Spiral Assume w.l.o.g. that p is the origin. A curve that forms a constant angle  with the radii is a logarithmic spiral, given by the polar equation (where b = cot -1  ):

The Optimal Backbone Near a Single Point Obstacle Proposition: Given a single point obstacle located at the origin, a start position and a goal position, where r s, r g  w max, the backbone of the optimal corridor connecting s and g is an arc of the logarithmic spiral, where: If r 0 = r s = r g then we have: a* = r 0, b* = 0 (a circle)

A Single Point Obstacle: The General Case Proposition: We are given two points s and g lying outside the disc B(p; w max ), where p is a point obstacle: If the segment sg does not intersect B(p; w max ), it is the optimal backbone. Otherwise, the optimal backbone comprises the tangent ss *, the circular arc s * g * and the tangent g * g. w max p g s s’ g’ s*s* g*g* w max p

Given a set O = {p 1, …, p n } of point obstacles in the plane that are well-separated with respect to w max, we use the visibility graph of the discs B(p 1 ; w max ), …, B(p n ; w max ) to compute optimal corridors. This graph can be constructed in O(n log n + E) time, where E is the number of visibility edges. The Visibility Graph Proposition: It is possible to compute the optimal corridor connecting s and g amidst n well-separated point obstacles in O(E log n) time.

The bounded Voronoi diagram Û ( O ) of a point set O w.r.t. w max comprises circular arcs (portions of the B(p i ; w max )’s) and line segments (portions of the Voronoi diagram). It subdivides the plane into: bounded Voronoi cells of obstacles, cells with clearance above w max. The Bounded Voronoi Diagram

In a cell whose clearance is above w max, optimal backbone paths comprise line segments and circular arcs. The weighted length of each element equals its “normal” length. Optimal Paths within Cells In the bounded Voronoi cell of p i the optimal backbone path between and is a spiral arc a whose weighted length is given by: Voronoi edges are also locally optimal.

We have to consider spiral arcs that shortcut sharp turns (Voronoi vertices and reflex vertices): Moving amidst Dense Obstacles pipi pipi pipi pjpj v v1v1 v2v2 v1v1 v2v2 q1q1 q2q2 q1q1 q2q2 q1q1 q2q2 q’q’ In the worst case we have to consider shortcuts of whose complexity is  (n), which take exponential time to compute.

Given  > 0, we subdivide the features of Û ( O ) into small intervals of length c(I)  / w max. (c(I) is the clearance of the interval I, which we assume to be constant as I is small). The Approximation Scheme (I) Let  be the total weighted length of all the features of Û ( O ), then there are  /  intervals in total. We define a graph D over the set of  /  intervals, which forms the node set of the graph.

D contains an edge between I 1 and I 2 only if the intervals are incident to a common cell. Such an edge may be: a straight visibility segment, a circular arc, a spiral arc, a straight line segment on a Voronoi edge. An edge must not cross a feature of Û ( O ). The Approximation Scheme (II) There are O(  2 /  2 ) edges in the graph, which can be constructed in O(n  2 /  2 ) time.

Given two endpoints, we can connect them to the graph D in O(n  /  ) time and use Dijkstra’s algorithm to obtain an approximation  for the optimal backbone path  * in O(  2 /  2 ) time. The Approximation Scheme (III) Lemma: If  * comprises the path segments  1, …,  k (where k = O(n)), then for each  i there exists an edge e in D such that: L * (e)  L * (  i ) + 2  2 . Corollary: We can compute in polynomial time a backbone path whose weighted length satisfies: L * (  )  L * (  * ) + O(n) .

Let us consider a scene cluttered with polygonal obstacles P = {P 1, …, P k } having n vertices in total. Polygonal Obstacles If the polygons are well-separated w.r.t. w max, we can use the visibility graph of the dilated obstacles to answer corridor- planning queries.

The features of Û ( P ) also include: line segments on dilated obstacle boundaries, parabolic arcs of the Voronoi diagram, line segments that separate bounded Voronoi cells of a polygon vertex from the cell of an incident edge. The Bounded Voronoi Diagram for Polygons

Using Snell’s Law: Moving Near a Polygon Edge (I) 00 p 0 = (x 0,0) x = 0 11 22 33 Note that we may have a reflection point!

Moving Near a Polygon Edge (II) Let us now write: Let  x  0: We obtained an equation of a circle whose center lies on x = 0.

The Optimal Backbone Near a Polygon Edge Proposition: The optimal backbone path between s and g near a polygon edge supported by x = 0 is a circular arc a of radius r * centered at (0, y * ), where: The weighted length of this arc is given by ( and w.r.t. the circle center):

We can generalize our approximation scheme for the case of polygonal obstacles. For any  > 0, we can construct a graph D based on the structure of Û ( P ) in O(n  2 /  2 ) time (w.r.t. a given w max value). Optimal Corridors amidst Polygons For any two given endpoints, we can query D in O(  2 /  2 ) time and obtain a path that approximates the best backbone path up to an error of O(n) .

Thank you!