Autonomous Robots Robot Path Planning (2) © Manfred Huber 2008

Path Planning Basic Motion Planning Problem in Configuration Space Solid robot reduced to a single point (by extending obstacles appropriately) Only static obstacles Holonomic robot Only collision free paths are allowed This is just for a small recap of the material covered in the previous class © Manfred Huber 2008

Roadmap Approaches Construct a set of roads and determine the path by finding a sequence of roads that lead from the start to the goal. Visibility graphs Pros: Complete, Correct, Optimal (in terms of distance Cons: Paths very close to obstacles and with sharp corners, very hard to construct beyond 2D Voronoi diagrams Pros: Complete, Correct, relatively safe and smooth path Cons: Harder to construct roads, potentially long paths These were covered last class © Manfred Huber 2008

Cell Decomposition Approaches Autonomous Robots Robot Path Planning: Cell Decomposition Approaches Next type of path planners: One of the main motivations for these is to further reduce the complexity of path planning and avoid the problems of constructing complex road networks in higher dimensional spaces. © Manfred Huber 2008

Cell Decomposition Decompose the workspace into freespace and obstacle cells. Within freespace cells the robot can move freely Path planning reduces to: Finding (searching for) a sequence of adjacent freespace cells with the first containing the current location and the final one containing the goal Determining a strategy to move within and between freespace cells © Manfred Huber 2008

Moving Within and Between Cells Most cell decomposition approaches rely on the construction of convex polygonal cells Convex cells allow for safe motion between any two points in a cell along a straight line. Polygonal cells allow for safe transition between cells by moving through the midpoint of the shared edge between consecutive cells on a path. © Manfred Huber 2008

Cell Decomposition Cell Construction Path Search Path Construction Construct convex freespace cells 1 2 3 4 5 6 Path Search Label cells and search for path 5 3 6 2 4 General example of cell decomposition path planning The main difference between different cell decomposition techniques is n the way in which freespace cells are constructed. 1 Path Construction Connect centers of cell connections © Manfred Huber 2008

Cell Decomposition Complete ? Correct ? Optimal ? For exact cell decomposition (i.e. if every point in free space is part of a freespace cell): Yes Correct ? If cells are convex and no part of an obstacle is in a freespace cell: Yes Optimal ? No © Manfred Huber 2008

Exact Cell Decomposition Exact cell decomposition requires: Freespace cells contain only freespace No risk of encountering obstacles when moving within a cell Obstacle cells contain only obstacles No piece of freespace is inaccessible to the robot. © Manfred Huber 2008

General Convex Polygonal Decoposition Construct polygonal convex cells © Manfred Huber 2008

General Convex Polygonal Decomposition Advantages Small number of cells Fast path search. Disadvantages: Cells very hard to construct automatically due to complex geometry © Manfred Huber 2008

Trapezoidal Decoposition Construct freespace cells as trapezoids If limiting us to trapezoids that have two parallel vertical sides, a trapezoidal decomposition is relatively easy to form by simply going from left to right and decomposing space with a vertical line every time a new obstacle corner occurs. © Manfred Huber 2008

Trapezoidal Decomposition Advantages Cells are easy to construct Very systematic and algorithmic construction phase Disadvantages: Larger number of cells than for general convex decomposition © Manfred Huber 2008

Approximate Cell Decomposition Freespace cells contain only freespace No risk of encountering obstacles when within a cell Obstacle cells contain only obstacles No piece of freespace is inaccessible to the robot. Mixed cells contain both obstacles and freespace Have to be treated like obstacle cells to ensure correctness of the path Approximate cell decomposition is not complete since a valid path could lead through a mixed cell. To allow for even faster construction of cell decompositions, even simpler geometries of freespace cells (such as rectangles or squares) would be useful. But that prevents covering all of freespace. Approximate cell decomposition approaches address this by trading off ease of construction for completeness of the path planner. © Manfred Huber 2008

General Approximate Decoposition Divide space into (mostly) rectangular regions © Manfred Huber 2008

Approximate Cell Decomposition Common approximate decompositions: Regular grids Binary or quad trees Bounding rectangles © Manfred Huber 2008

Regular Grid Decomposition Divide space into a regular grid and mark cells that contain any piece of an obstacle © Manfred Huber 2008

Regular Grid Decomposition Advantages: Very easy to construct Easy to find cell sequence as each has the same number of neighbors Easy to construct path since edges between cells are predefined Disadvantages: Very large number of cells To allow for even faster construction of cell decompositions, even simpler geometries of freespace cells (such as rectangles or squares) would be useful. But that prevents covering all of freespace. Approximate cell decomposition approaches address this by trading off ease of construction for completeness of the path planner. © Manfred Huber 2008

Binary Tree Decomposition Divide space by iteratively halving mixed cells vertically or horizontally until a size limit © Manfred Huber 2008

Binary Tree Decomposition Advantages: Easy to construct Smaller number of cells compared to regular grid at same resolution Disadvantages: Higher construction effort as for regular grid To allow for even faster construction of cell decompositions, even simpler geometries of freespace cells (such as rectangles or squares) would be useful. But that prevents covering all of freespace. Approximate cell decomposition approaches address this by trading off ease of construction for completeness of the path planner. © Manfred Huber 2008

Quad Tree Decomposition Divide space by iteratively cutting mixed cells into 4 pieces until a size limit © Manfred Huber 2008

Quad Tree Decomposition Advantages: Easy to construct Slightly faster than binary decomposition Disadvantages: More cells than binary decomposition at same resolution To allow for even faster construction of cell decompositions, even simpler geometries of freespace cells (such as rectangles or squares) would be useful. But that prevents covering all of freespace. Approximate cell decomposition approaches address this by trading off ease of construction for completeness of the path planner. © Manfred Huber 2008

Bounding Rectangles Enclose freespace in bounded rectangles until mixture percentage limit. © Manfred Huber 2008

Bounding Rectangles Advantages: Disadvantages: Very small number of cells Better coverage than other rectangle-based approaches Disadvantages: Harder to construct automatically To allow for even faster construction of cell decompositions, even simpler geometries of freespace cells (such as rectangles or squares) would be useful. But that prevents covering all of freespace. Approximate cell decomposition approaches address this by trading off ease of construction for completeness of the path planner. © Manfred Huber 2008

Approximate Cell Decomposition Advantages: Faster to construct than exact decomposition Easier to represent in data structures since cells are rectangular Disadvantages: Complete only to the resolution of the cells: If no path with sufficient clearance (at least two times the diagonal of the cells) exists, no path might be found. To allow for even faster construction of cell decompositions, even simpler geometries of freespace cells (such as rectangles or squares) would be useful. But that prevents covering all of freespace. Approximate cell decomposition approaches address this by trading off ease of construction for completeness of the path planner. © Manfred Huber 2008

Cell Decomposition Advantages: Disadvantages: Often faster to compute than roadmap based techniques Easier to expand into higher dimensional configuration space than roadmap approaches Cells are easier to represent than roads Disadvantages: Exact decomposition can be complex Approximate decomposition is not complete To allow for even faster construction of cell decompositions, even simpler geometries of freespace cells (such as rectangles or squares) would be useful. But that prevents covering all of freespace. Approximate cell decomposition approaches address this by trading off ease of construction for completeness of the path planner. © Manfred Huber 2008