Jane Li - WPIusers.wpi.edu/~zli11/teaching/rbe550_2018/Lecture... · h1(s) only h1(s) and h2(s) RBE...
Transcript of Jane Li - WPIusers.wpi.edu/~zli11/teaching/rbe550_2018/Lecture... · h1(s) only h1(s) and h2(s) RBE...
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Jane Li
Assistant Professor
Mechanical Engineering Department, Robotic Engineering Program
Worcester Polytechnic Institute
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• A search-algorithm prioritizes and expands the nodes in its open list items by their key values. How to compute the key values for
• (2 pts) Dijkstra algorithm
• (2 pts) A* algorithm
• (3 pts) ARA*
• (3 pts) ANA*
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Dijkstra𝑓 𝑠 = 𝑔 𝑠
Optimality of Path to Goal
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A*
𝑓 𝑠 = 𝑔 𝑠 + ℎ(𝑠)
• Converges Quickly to Optimal Path
• Heuristic Functions
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ARA*
• Issue WA*: Terminates with Sub-Optimal Solution
• ARA*: Iteratively reduces 𝜖 till Optimal Solution
𝑓 𝑠 = 𝑔 𝑠 + 𝜖 ⋅ ℎ 𝑠𝜖𝑛𝑒𝑤 = 𝜖𝑜𝑙𝑑 − Δ𝜖
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ANA* 𝑓 𝑠 = 𝑔 𝑠 + 𝜖 ⋅ ℎ(𝑠)
• No Parameters
• Elegant choice of Δ𝜖 => Selected On-the-fly
𝑒 𝑠 =𝐺 − 𝑔(𝑠)
ℎ(𝑠)𝜖𝑛𝑒𝑤 = max
𝑠∈𝑂𝑃𝐸𝑁𝑒(𝑠)
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• More on search algorithms• Toward optimal and quicker solutions
• Variants of A*
• Practical issues• Case study – Practical search techniques for autonomous driving
• Other advanced topics • Roadmaps for planning in dynamic environment
• Learning search via imitation
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• Task
• Autonomous driving in unknown environment
• Re-plan online while incrementally building an obstacle map
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Stanford Racing TeamJunior in Urban Challenge (DARPA 2007)
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• Capability – complex general path-planning tasks
• Navigating parking lots
• Executing U-turns
• Dealing with blocked roads and intersections
• Performance
• Typical full-cycle re-planning times of 50–300ms on a modern PC
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• Robot control and trajectories must be continuous
• Continuous-variable optimization problem
• Existing search algorithms are fast in discrete state space
• Produce non-smooth paths
• Do not satisfy vehicle’s non-holonomic constraints
• Other approaches?
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• Forward search in continuous coordinates
• Guarantees kinematic feasibility
• Need an efficient heuristic to guide expansion
• Non-linear optimization in control space
• Fast convergence is difficult due to local minima
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• Phase 1
• Heuristic search in continuous coordinates to find a kinematicallyfeasible solution do not care optimality
• Phase 2
• Use gradient descent to find at least the locally optimal solution
• Global optimal is possible
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• Compared to A*
• Search space
• Associates with each grid cell a continuous 3D state of the vehicle
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A* Field D* Hybrid-state A*
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• Features
• Associates costs with cell centers
• Only visit states at cell centers
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• Features
• Associate costs with cell corners
• Allow arbitrary linear paths from cell to cell
• [Ferguson and Stentz 2005]
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• Features
• Associate a continuous state with each cell
• Score of the cell = the cost of its associated continuous state
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• Guaranteed to find the minimal-cost?
• No
• Guarantee to be drivable?
• Yes
• Sub-optimality w.r.t. global optimal solution?
• In practice, hybrid-A* is close to global optimal solution refine in Phase 2
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• Can maneuver in tight spaces?
• Like what?
• Plan forward and reverse motion?
• Yes
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• Which nodes to be expanded first
• Bad heuristics leads to wasted expansion
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• h1(s) = Non-holonomic without obstacles
• Ignores obstacles
• Consider non-holonomic constraints
• Admissible?
• Use the max of h1(s) and Euclidian distance – Benefits?
• Prune search branches that approach the goal with the wrong headings
• Computed offline
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• h2(s) = dual of h1(s)
• Consider obstacles
• Ignore non-holonomic constraints
• Admissible?
• Benefit?
• Discover all U-shaped obstacles dead-ends in 2D
• Guide the more expensive 3D search away from these areas
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Euclidian Distance h1(s)
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h1(s) only h1(s) and h2(s)
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How to precisely determine where these points are?
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• Driving motion primitives
• A node in the tree is expanded by simulating a kinematic model of the car (using a particular control action) for a small period of time
• The resulting Reed-Shepp path is then checked for collision against the current obstacle map
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• Yellow-green path
• Search tree branches (i.e., short incremental expansions)
• Pink path
• Reed-Shepp path leading towards the goal
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• Design a cost function that yields the desired driving behavior
• Criteria??
• Penalizing proximity to obstacles Potential field
• Create high-potential areas in narrow passages cannot pass
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• Rescale the field based on the workspace geometry
• Otherwise,
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• Rescale the field based on the workspace geometry
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• Improve path length and solution smoothness
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Prefer clearance (Voronoi Field) Penalize collisions with obstacles
Bound trajectory curvature Measure of path smoothness
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• Improve path resolution by interpolation
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• [1] Dolgov, Dmitri, Sebastian Thrun, Michael Montemerlo, and James Diebel. "Practical search techniques in path planning for autonomous driving." Ann Arbor 1001, no. 48105 (2008): 18-80.
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