10 chapter 10: metric path planning a. representations b. algorithms
TRANSCRIPT
10Chapter 10:
Metric Path Planning
a. Representationsb. Algorithms
Chapter 10: Metric Path Planning 2
10 Representing Area/Volume in Path Planning
• Quantitative or metric– Rep: Many different ways to represent an area
or volume of space• Looks like a “bird’s eye” view, position &
viewpoint independent– Algorithms
• Graph or network algorithms• Wavefront or graphics-derived algorithms
Chapter 10: Metric Path Planning 3
10 Metric Maps• Motivation for having a metric map is often path
planning (others include reasoning about space…)
• Determine a path from one point to goal– Generally interested in “best” or “optimal” What are measures
of best/optimal?
– Relevant: occupied or empty
• Path planning assumes an a priori map of relevant aspects– Only as good as last time map was updated
Chapter 10: Metric Path Planning 4
10 Metric Maps use Cspace• World Space: physical space robots and obstacles existin
– In order to use, generally need to know (x,y,z) plus Euler angles: 6DOF
• Ex. Travel by car, what to do next depends on where you are and what direction you’re currently heading
• Configuration Space (Cspace)– Transform space into a representation suitable for robots,
simplifying assumptions
6DOF 3DOF
Chapter 10: Metric Path Planning 5
10 Major Cspace Representations
• Idea: reduce physical space to a cspace representation which is more amenable for storage in computers and for rapid execution of algorithms
• Major types
– Meadow Maps
– Generalized Voronoi Graphs (GVG)
– Regular grids, quadtrees
Chapter 10: Metric Path Planning 6
10 Meadow Maps
• Example of the basic procedure of transforming world space to cspace
• Step 1 (optional): grow obstacles as big as robot
Chapter 10: Metric Path Planning 7
10 Meadow Maps cont.• Step 2: Construct convex polygons as line segments between pairs
of corners, edges– Why convex polygons? Interior has no obstacles so can safely transit
(“freeway”, “free space”)
– Oops, not necessarily unique set of polygons
Chapter 10: Metric Path Planning 8
10 Meadow Maps cont.• Step 3: represent convex polygons in way suitable for path
planning-convert to a relational graph– Is this less storage, data points than a pixel-by-pixel representation?
Chapter 10: Metric Path Planning 9
10 Problems with Meadow Maps
• Not unique generation of polygons
• Could you actually create this type of map with sensor data?
• How does it tie into actually navigating the path?– How does robot recognize “right” corners, edges and go to
“middle”?
– What about sensor noise?
Chapter 10: Metric Path Planning 10
10 Path Relaxation
• Get the kinks out of the path– Can be used with any cspace representation
Chapter 10: Metric Path Planning 11
10 Generalized Voronoi Graphs • Create lines equidistant from objects and walls,
• Intersections of lines are nodes
• Result is a relational graph
Chapter 10: Metric Path Planning 12
10 Regular Grids
• Bigger than pixels, but same idea– Often on order of 4inches square
– Make a relational graph by each element as a node, connecting neighbors (4-connected, 8-connected)
– Moore’s law effect: fast processors, cheap hard drives, who cares about overhead anymore?
Chapter 10: Metric Path Planning 13
10 Problems with GVG and Regular Grids
• GVG– Sensitive to sensor noise
– Path execution requires robot to be able to sense boundaries
• Grids– World doesn’t always line up on grids
– Digitalization bias: left over space marked as occupied
Chapter 10: Metric Path Planning 14
10 Summary
• Metric path planning requires – Representation of world space, usually try to simplify to
cspace– Algorithms which can operate over representation to produce
best/optimal path
• Representation– Usually try to end up with relational graph– Regular grids are currently most popular in practice, GVGs
are interesting– Tricks of the trade
• Grow obstacles to size of robot to be able to treat holonomic robots as point
• Relaxation (string tightening)
• Metric methods often ignore issue of– how to execute a planned path– Impact of sensor noise or uncertainty, localization
Chapter 10: Metric Path Planning 15
10 Algorithms
• Path planning– A* for relational graphs
– Wavefront for operating directly on regular grids
• Interleaving Path Planning and Execution
Chapter 10: Metric Path Planning 16
10 Motivation for A*
• Single Source Shortest Path algorithms are exhaustive, visting all edges– Can’t we throw away paths when we see that they aren’t going
to the goal, rather than follow all branches?
• This means having a mechanism to “prune” branches as we go, rather than after full exploration
• Algorithms which prune earlier (but correctly) are preferred over algorithms which do it later.
• Issue: the mechanism for pruning
Chapter 10: Metric Path Planning 17
10 A*• Similar to breadth-first: at each point of time the
planner can only “see” it’s node and 1 set of nodes “in front”
• Idea is to rate the choices, choose the best one first, throw away any choices whenever you can
• f*(n) is the “cost” of the path from Start to Goal through node n
• g*(n) is the “cost” of going from Start to node n• h*(n) is the cost of going from n to the Goal
– h* is a “heuristic function” because it must have a way of guessing the cost of n to Goal since it can’t see the path between n and the Goal
f*(n)=g*(n)+h*(n)
Chapter 10: Metric Path Planning 18
10 A* Heuristic Function
• g*(n) is easy: just sum up the path costs to n
• h*(n) is tricky– path planning requires an a priori map
– Metric path planning requires a METRIC a priori map
– Therefore, know the distance between Initial and Goal nodes, just not the optimal way to get there
– h*(n)= distance between n and Goal
– h*(n) <= h(n)
f*(n)=g*(n)+h*(n)
Chapter 10: Metric Path Planning 19
10 Example: A to E
• But since you’re starting at A and can only look 1 node ahead, this is what you see:
AB
D
F E
1
1
1
1
1.4
1.4
AB
D
E
1
1.4
Chapter 10: Metric Path Planning 20
10
• Two choices for n: B, D
• Do both– f*(B)=1+2.24=3.24
– f*(D)=1.4+1.4=2.8
• Can’t prune, so much keep going (recurse)– Pick the most plausible path first A-D-?-E
AB
D
E
1
1.4
1.4
2.24
Chapter 10: Metric Path Planning 21
10
• A-D-?-E– “stand on D”– Can see 2 new nodes: F, E– f*(F)=(1.4+1)+1=3.4– f*(E)=(1.4+1.4)+0=2.8
• Three paths– A-B-?-E >= 3.24– A-D-E = 2.8– A-D-F-?-D >=3.4
• A-D-E is the winner! – Don’t have to look farther because expanded the shortest first, others couldn’t
possibly do better without having negative distances, violations of laws of geometry…
AB
D
E
1
1.4
1.4
F1
1
Chapter 10: Metric Path Planning 22
10 Wavefront Planners
Chapter 10: Metric Path Planning 23
10 Trulla
Chapter 10: Metric Path Planning 24
10 Interleaving Path Planning and Reactive Execution
• Graph-based planners generate a path and subpaths or subsegments
• Recall NHC, AuRA– Pilot looks at current subpath, instantiates behaviors to get from
current location to subgoal
• When the robot tries to reach a subgoal, it may exhibit subgoal obsession due to an encoder error - it is necessary to allow a tolerance corresponding usually to +/- width of robot
• What happens if a goal is blocked? - need a Termination condition, e.g. deadline
• What happens if a robot avoiding an obstacle is now closer to the next subgoal? - it would be good to use an opportunistic replanning
Chapter 10: Metric Path Planning 25
10 Two Example Approaches
• If computing all possible paths in advance, there not a problem– Shortest path between pairs will be part of
shortest path to more distant pairs
• D* – Run A* over all possible pairs of nodes
– continuously update the map
– disadvantages: 1. too computationally expensive to be practical for a robot 2. continuous replanning is highly dependent on sensing quality,
Chapter 10: Metric Path Planning 26
10 Two Example Approaches– Event driven scheme - event noticeable by
a reactive system would trigger replanning
– the Trulla planner uses for this dot product of the intended path vector and the actual path vector
– By-product of wave propagation style is path to everywhere
– for opportunistic replanning in case of favorable change D* is better, because it will automatically notice the change, while Trulla will not notice it
Chapter 10: Metric Path Planning 27
10 Trulla Example
Chapter 10: Metric Path Planning 28
10 Summary
• Metric path planners– graph-based (A* is best known)
– Wavefront
• Graph-based generate paths and subgoals.– Good for NHC styles of control
– In practice leads to:• Subgoal obsession
• Termination conditions
• Planning all possible paths helps with subgoal obsession– What happens when the map is wrong, things change, missed
opportunities? How can you tell when the map is wrong or that’s it worth the computation?
Chapter 10: Metric Path Planning 29
10 You should be able to:
• Define Cspace, path relaxation, digitization bias, subgoal obsession, termination condition
• Explain the difference between graph and wavefront planners
• Represent an indoor environment with a GVG, a regular grid, or a quadtree, and create a graph suitable for path planning
• Apply A* search
• Apply wavefront propagation
• Explain the differences between continuous and event-driven replanning