introduction to mobile robotics path planning and collision...
TRANSCRIPT
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Path Planning and Collision Avoidance
Introduction to Mobile Robotics
Wolfram Burgard, Maren Bennewitz,
Diego Tipaldi, Luciano Spinello
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Motion Planning
Latombe (1991): “… is eminently necessary since, by definition, a robot accomplishes tasks by moving in the real world.”
Goals: § Collision-free trajectories § Robot should reach the goal location as
quickly as possible
Optimality
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… in Dynamic Environments § How to react to unforeseen obstacles?
§ efficiency § reliability
§ Dynamic Window Approaches [Simmons, 96], [Fox et al., 97], [Brock & Khatib, 99]
§ Grid map based planning [Konolige, 00]
§ Nearness Diagram Navigation [Minguez at al., 2001, 2002]
§ Vector-Field-Histogram+ [Ulrich & Borenstein, 98]
§ A*, D*, D* Lite, ARA*, …
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Two Challenges § Calculate the optimal path taking potential
uncertainties in the actions into account
§ Quickly generate actions in the case of unforeseen objects
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Classic Two-Layered Architecture
Planning
Collision Avoidance
sensor data
map
robot
low frequency
high frequency
sub-goal
motion command
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Dynamic Window Approach
§ Collision avoidance: Determine collision-free trajectories using geometric operations
§ Here: Robot moves on circular arcs § Motion commands (v,ω) § Which (v,ω) are admissible and reachable?
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Admissible Velocities § A speed is admissible if the robot is able to
stop before colliding with an obstacle
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Reachable Velocities § Speeds that are reachable by acceleration
within the time period t:
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DWA Search Space
§ Vs = all possible speeds of the robot § Va = obstacle free area § Vd = speeds reachable within a certain time frame based on
possible accelerations
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Dynamic Window Approach
§ How to choose <v,ω>? § Steering commands are chosen by a
heuristic navigation function
§ This function tries to minimize the travel-time by: “driving fast in the right direction”
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Dynamic Window Approach § Heuristic navigation function § Planning restricted to <x,y>-space § No planning in the velocity space
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβαNavigation Function: [Brock & Khatib, 99]
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goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβαNavigation Function: [Brock & Khatib, 99]
Maximizes velocity.
§ Heuristic navigation function § Planning restricted to <x,y>-space § No planning in the velocity space
Dynamic Window Approach
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goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβαNavigation Function: [Brock & Khatib, 99]
Considers cost to reach the goal.
Maximizes velocity.
§ Heuristic navigation function § Planning restricted to <x,y>-space § No planning in the velocity space
Dynamic Window Approach
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goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβαNavigation Function: [Brock & Khatib, 99]
Maximizes velocity.
Considers cost to reach the goal.
Follows grid based path computed by A*.
§ Heuristic navigation function § Planning restricted to <x,y>-space § No planning in the velocity space
Dynamic Window Approach
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Navigation Function: [Brock & Khatib, 99] Goal nearness.
Follows grid based path computed by A*.
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβαMaximizes velocity.
Considers cost to reach the goal.
§ Heuristic navigation function § Planning restricted to <x,y>-space § No planning in the velocity space
Dynamic Window Approach
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Dynamic Window Approach § Reacts quickly § Low CPU power requirements § Guides a robot on a collision-free path § Successfully used in a lot of real-world
scenarios
§ Resulting trajectories sometimes sub-optimal
§ Local minima might prevent the robot from reaching the goal location
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
Robot‘s velocity.
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
Preferred direction of NF.
Robot‘s velocity.
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
§ The robot drives too fast at c0 to enter the corridor facing south.
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
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Problems of DWAs
goalnfnfvelNF ⋅+Δ⋅+⋅+⋅= δγβα
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Problems of DWAs
§ Same situation as in the beginning
è DWAs might not be able to reach the goal location.
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Problems of DWAs § Typical problem in a real world situation:
§ Robot does not slow down early enough to enter the doorway
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Robot Path Planning with A* § Finds the shortest path
§ Requires a graph structure
§ Limited number of edges
§ In robotics: Often planning using a 2D occupancy grid map
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Reminder: A*
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§ g(n) = actual cost from the initial state to n
§ h(n) = estimated cost from n to the next goal
§ f(n) = g(n) + h(n), the estimated cost of the cheapest solution through n
§ Let h*(n) be the actual cost of the optimal path from n to the next goal
§ h is admissible if the following holds for all n :
h(n) ≤ h*(n)
§ A* yields the optimal path if h is admissible (the straight-line distance is admissible in the Euclidean Space)
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Example: Path Planning with A* for Robots in a Grid-World
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Deterministic Value Iteration § To compute the shortest path from
every state to one goal state, use (deterministic) value iteration
§ Very similar to Dijkstra’s Algorithm
§ Such a cost distribution is the optimal heuristic for A*
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Typical Assumption in Robotics for A* Path Planning
§ The robot is assumed to be localized § The robot computes its path based on
an occupancy grid § Motion commands are executed
accurately
Is this always true?
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Problems
§ What if the robot is slightly delocalized?
§ Moving on the shortest path guides often the robot on a trajectory close to obstacles
§ Trajectory aligned to the grid structure
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Convolution of the Grid Map
§ Convolution blurs the map
§ Obstacles are assumed to be bigger than in reality
§ Perform an A* search in such a convolved map
§ Robot increases distance to obstacles and moves on a short path!
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Example: Map Convolution
§ 1-d environment, cells c0, …, c5
§ Cells before and after 2 convolution runs
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Convolution § Consider an occupancy map. The
convolution is defined as:
§ This is done for each row and each column of the map
§ “Gaussian blur”
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A* in Convolved Maps
§ The costs are a product of path length and occupancy probability of the cells
§ Cells with higher probability (e.g., caused by convolution) are avoided by the robot
§ Thus, it keeps distance to obstacles
§ This technique is fast and quite reliable
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5D-Planning – An Alternative to the Two-layered Architecture
§ Plans in the full <x,y,θ,v,ω>-configuration space using A*
è Considers the robot's kinematic constraints
§ Generates a sequence of steering commands to reach the goal location
§ Maximizes trade-off between driving time and distance to obstacles
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The Search Space (1)
§ What is a state in this space? <x,y,θ,v,ω> = current position and
velocities of the robot § How does a state transition look like?
<x1,y1,θ1,v1,ω1> <x2,y2,θ2,v2,ω2> with motion command (v2,ω2) and |v1-v2| < av, |ω1-ω2| < aω
§ Pose of the robot is a result of the motion equations
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The Search Space (2)
Idea: Search in the discretized <x,y,θ,v,ω>-space
Problem: The search space is too huge to
be explored within the time constraints (5+ Hz for online motion plannig)
Solution: Restrict the full search space
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The Main Steps of Our Algorithm
1. Update (static) grid map based on sensory input
2. Use A* to find a trajectory in the <x,y>-space using the updated grid map
3. Determine a restricted 5d-configuration space based on step 2
4. Find a trajectory by planning in the restricted <x,y,θ,v,ω>-space
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Updating the Grid Map
§ The environment is represented as a 2d-occupency grid map
§ Convolution of the map increases security distance
§ Detected obstacles are added § Cells discovered free are cleared
update
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Find a Path in the 2d-Space
§ Use A* to search for the optimal path in the 2d-grid map
§ Use heuristic based on a deterministic value iteration within the static map
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Restricting the Search Space
Assumption: The projection of the optimal 5d-path onto the <x,y>-space lies close to the optimal 2d-path
Therefore: Construct a restricted search
space (channel) based on the 2d-path
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Space Restriction
§ Resulting search space = <x, y, θ, v, ω> with (x,y) Є channel
§ Choose a sub-goal lying on the 2d-path within the channel
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Find a Path in the 5d-Space § Use A* in the restricted 5d-space to find a
sequence of steering commands to reach the sub-goal
§ To estimate cell costs: Perform a
deterministic 2d-value iteration within the channel
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Examples
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Timeouts
§ Steering a robot online requires to set a new steering command every .25 secs
è Abort search after .25 secs.
How to find an admissible steering command?
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Alternative Steering Command
§ Previous trajectory still admissible? è OK
§ If not, drive on the 2d-path or use
DWA to find new command
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Timeout Avoidance
è Reduce the size of the channel if the 2d-path has high cost
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Example
Robot Albert Planning state
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Comparison to the DWA (1)
§ DWAs often have problems entering narrow passages
DWA planned path. 5D approach.
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Comparison to the DWA (2)
The presented approach results in significantly faster motion when driving through narrow passages!
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Comparison to the Optimum
Channel: with length=5m, width=1.1m Resulting actions are close to the optimal solution
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Summary § Robust navigation requires combined path
planning & collision avoidance § Approaches need to consider robot's kinematic
constraints and plans in the velocity space § Combination of search and reactive techniques
show better results than the pure DWA in a variety of situations
§ Using the 5D-approach the quality of the trajectory scales with the computational resources available
§ The resulting paths are often close to the optimal ones
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What’s Missing?
§ More complex vehicles (e.g., cars).
§ Moving obstacles, motion prediction.
§ Path planning
§ …