Motion Planning & Robot Planning

Motion Planning & Robot Planning

Prof.: S. ShiryMohsen gandomkar

M.Sc Artificial IntelligenceDepartment of Computer Eng. and IT

Amirkabir Univ. of Technology(Tehran Polytechnic)

What is Motion Planning?

• Motion planning is aimed at providing robots with the capability of deciding automatically which motions to execute in order to achieve their tasks without colliding with other objects in their work space

Basic Definition

• Obstacles• Already occupied spaces of the world• In other words, robots can’t go there

• Free Space• Unoccupied space within the world• Robots “might” be able to go here• To determine where a robot can go, we need to

discuss what a Configuration Space is

For a point robot moving in 2-D plane, C-space is

Configuration Space of A is the space (C) of all possible configurations of A.

C Cfree

Cobs

2R

The Configuration Space

qstart

qgoal

For a point robot moving in 3-D, the C-space is

x

y

qstart

qgoal

C

Cfree

Cobs

3R

What is the difference between Euclidean space and C-space?



• How to create it• First abstract the robot as a point object.

Then, enlarge the obstacles to account for the robot’s footprint and degrees of freedom

• In our example, the robot was circular, so we simply enlarged our obstacles by the robot’s radius (note the curved vertices)

Example of a World (and Robot)

Obstacles

Free Space

Robot

x,y

Configuration Space: Accommodate Robot Size

Obstacles

Free Space

Robot(treat as point object)x,y

Motion Planning

• Basic problem: Collision-free path planning for one rigid or articulated object (the “robot”) among static obstacles.

• Inputs• geometric descriptions of the obstacles and the robot• kinematic and dynamic properties of the robot• initial and goal positions (configurations) of the robot

• Output• Continuous sequence of collision-free configurations

connecting the initial and goal configurations.

• Complete Algorithms• Probabilistic Algorithms• Heuristic Algorithms

Algorithmic Approaches

• Guaranteed to find a free path between two give configurations when exists and report failure otherwise

• Deal with connectivity of free space by capturing it on a graph.• Cell Decomposition - partition of free space• Roadmap Technique - network of curves

Complete Algorithms

• Trade-off exactness against running time• Don’t guarantee a solution but if exists very likely

to find it relatively quickly• Example: Probabilistic Roadmap Algorithm

• Experimental results show that computation takes less than a second

Probabilistic Algorithms

Heuristic Algorithms

• Many work well in practice but offer no performance guarantee

• Deal with a grid on configuration space• Example 1 : Potential Field• Example 2 : Approximate Cell Decomposition

Previous Approaches

Visibility Graphs

Voronoi Diagrams

Exact Cell Decomposition

Approximate Cell Decomposition

Potential Fields

Probabilistic Roadmaps Method

Problems before PRMs

• Hard to plan for many dof robots• Computation complexity for high-dimensional

configuration spaces would grow exponentially• Potential fields run into local minima• Complete, general purpose algorithms are at best

exponential and have not been implemented

Difficulty with classic approaches

• Running time increases exponentially with the dimension of the configuration space.• For a d-dimension grid with 10 grid points on

each dimension, how many grid cells are there?

• Several variants of the path planning problem have been proven to be PSPACE-hard.

10d

Probabilistic Roadmap (PRM): multiple queries

free space

[Kavraki, Svetska, Latombe,Overmars, 96]

local path

milestone

Probabilistic Roadmap (PRM): single query

Multiple-Query PRM

Classic multiple-query PRM

• Probabilistic Roadmaps for Path Planning in High-Dimensional Configuration Spaces, L. Kavraki et al., 1996.

Assumptions

• Static obstacles• Many queries to be processed in the same

environment• Examples

• Navigation in static virtual environments• Robot manipulator arm in a workcell

Enter PRMs

• PRMs use fast collision checking techniques• PRMs avoid computing an explicit representation of the

configuration space • Two Phases

• A Learning Phase• A Query Phase

The Learning Phase

• Construct a probabilistic roadmap by generating random free configurations of the robot and connecting them using a simple, but very fast motion planer, also know as a local planner

• Store as a graph whose nodes are the configurations and whose edges are the paths computed by the local planner

PRM - Learning Phase

Free space

C-obstacle


Free space

C-obstacle

Free space

milestones

C-obstacle



Free space

milestones

C-obstacle

The Query Phase

• Find a path from the start and goal configurations to two nodes of the roadmap

• Search the graph to find a sequence of edges connecting those nodes in the roadmap

• Concatenating the successive segments gives a feasible path for the robot

Two geometric primitives in configuration space

• CLEAR(q)Is configuration q collision free or not?

• LINK(q, q’) Is the straight-line path between q and q’ collision-free?

Uniform sampling

Input: geometry of the moving object & obstaclesOutput: roadmap G = (V, E)

1: V and E .2: repeat3: q a configuration sampled uniformly at random from C.4: if CLEAR(q)then5: Add q to V.6: Nq a set of nodes in V that are close to q.

6: for each q’ Nq, in order of increasing d(q,q’)7: if LINK(q’,q)then8: Add an edge between q and q’ to E.

Difficulty

• Many small connected components

Resampling (expansion)

• Failure rate

• Weight

• Resampling probability

LINK no.

LINK failed no.)( qr

ppr

qrqw

)(

)()(

)()(Pr qwq

Resampling (expansion)

Query processing

• Connect qinit and qgoal to the roadmap

• Start at qinit and qgoal, perform a random walk, and try to connect with one of the milestones nearby

• Try multiple times

Error

• If a path is returned, the answer is always correct.• If no path is found, the answer may or may not be

correct. We hope it is correct with high probability.

Why does it work? Intuition

• A small number of milestones almost “cover” the entire configuration space.

Smoothing the path

Smoothing the path

Single-Query PRM

Lazy PRM

• Path Planning Using Lazy PRM, R. Bohlin & L. Kavraki, 2000.

Precomputation: roadmap construction

• Nodes• Randomly chosen configurations, which may or

may not be collision-free• No call to CLEAR

• Edges• an edge between two nodes if the

corresponding configurations are close according to a suitable metric

• no call to LINK

Query processing: overview

1. Find a shortest path in the roadmap

2. Check whether the nodes and edges in the path are collision.

3. If yes, then done. Otherwise, remove the nodes or edges in violation. Go to (1).

• We either find a collision-free path, or exhaust all paths in the roadmap and declare failure.

Query processing: details

• Find the shortest path in the roadmap• A* algorithm• Dijkstra’s algorithm

• Check whether nodes and edges are collisions free• CLEAR(q)• LINK(q0, q1)

Node enhancement

• Select nodes that close the boundary of F

Bug algorithms

Bug algorithms

• Assumptions:• Point robot• Contact sensor (Bug1,Bug2) or finite range sensor

(Tangent Bug)• Bounded environment• Robot position is perfectly known• Robot can measure the distance between two

points

Bug algorithms

• Algorithm consists of two behaviors:

1. Motion to goal – move toward the goal • Bug1: move along the line that connects an

“initial” point to the goal until you reach the goal or an obstacle (hit point).

• Bug2: move along the line that connects the start point to the goal until you reach the goal or an obstacle (hit point).

Bug algorithms

2. Boundary following – obstacle handeling • Bug1: circumnavigate the entire perimeter of the

obstacle, find the closest point to the goal on the perimeter (leave point), move to that point .

• Bug2: circumnavigate the obstacle until you reach a new point on the line connecting start and goal, that is closer to the goal (leave point).

Bug1 - example

qstart

qgoal

q1H

q1L

q2L

q2H

Motion to goal

Boundary following

Shortest path to goal

Bug2 - example

qstart

qgoal

q1H

Motion to goal

Boundary following

Line connecting start and goal

q1L

q2H

q2L

head-to-head comparison

What are worlds in which Bug 2 does better than Bug 1 (and vice versa) ?

Bug 2 beats Bug 1 Bug 1 beats Bug 2

Start

head-to-head comparison

What are worlds in which Bug 2 does better than Bug 1 (and vice versa) ?

Bug 2 beats Bug 1 Bug 1 beats Bug 2 “zipper world”

Start

Problem

Bug 2 beats Bug 1 Bug 1 beats Bug 2 “zipper world”

Problem

Adjusted bug algorithm Bug M1 • use Bug2 until the robot finds itself on the S-line farther from the goal than it started• if it does, revert to to Bug1 for that obstacle

Problem

Adjusted bug algorithm Bug M1 • use Bug2 until the robot finds itself on the S-line farther from the goal than it started• if it does, revert to to Bug1 for that obstacle

Bug1 vs. Bug2

Bug1 Exhaustive search Optimal leave point

Performs better with complex obstacles

Path length : n = # of obstacles P

i = perimeter of

obstacle i

Bug2 Opportunistic (greedy) search

Performs better with simple obstacles

Path length : n

i = # of times the

start-goal line intersects obstacle i

n

iigoalstartBug pqqdL

11 5.1),(

n

iiigoalstartBug pnqqdL

12 5.0),(

Finite range sensor

• Intervals of continuity

Tangent Bug algorithm

• Improvement to the Bug2 algorithm• Assumptions:

• All assumptions of Bug1/Bug2 except for contact sensor.

• Finite range sensor with 360◦ infinite orientation resolution.


• Like Bug1/Bug2, iterates between two behaviors: • motion to goal – consists of two parts:

• Move in a straight line towards the goal until you sense an obstacle directly between you and the goal

• Move toward an intermediate point* Oj according to some heuristic distance** until you reach the goal or until you reach a local minimum Mi in which case, switch to boundary following

• * Oj‘s are end points of an interval of continuity • ** For example d(x, Oj)+ d(Oj,goal)


Motion to goal

t = 1 t = 2 t = 3

o1

o2

t = 4

o1

o2

• Tangent Bug algorithm

• boundary following – define two distances:• dfollowing – The shortest distance between the sensed

boundary and the goal• dreach – The distance between the point on the boundary

that has a line of sight to the goal, and the goal

• continue moving around the obstacle in the same direction until dreach < dfollowing then switch to motion to goal


Boundary following

goalM

Motion to goal

Tangent Bug - example

qstart

qgoal

Motion to goal

Boundary following

Bug algorithms

• Simple and intuitive• Straightforward to implement• Success guaranteed (when possible)• Assumes perfect positioning and sensing • Sensor based planning – has to be incremental and

reactive

Multi-Robot Planning

Multi-Robot Planning Examples

Multi-Robot Planning

• An initial and a goal configuration are given as input for each robot

• Result is a coordinated path between the two configurations

• A coordinated path is one that indicates the configuration of every robot at each instant

• Collisions must be avoided between each pair of robot and obstacles, and between each pair of robots

Centralized Planning

• Paths for all robots are planned simultaneously by searching the c-space of the multi-arm robot

• Collisions between robots are self-collisions of the multi-arm robot

• For spot-welding example, 6 robots each with 6 dofs, so C will have 36-D

Centralized Planning

• Advantages• Complete – guaranteed to find a solution if one exists (if the

underlying planner is complete)

• Disadvantages• Potentially expensive – typically requires searching high-

dimensional spaces• Requires knowledge of goals and states of all robots

Decoupled Planning

• First Phase - a collision-free path ti is generated for each robot considering only obstacles (ignoring other robots) in its space

Decoupled Planning

• Second Phase (Velocity Tuning) – coordination of the robots’ velocities along their pre-generated paths to prevent collisions between robots. Two coordination methods discussed• Pairwise Coordination• Global Coordination

• Each robot is restricted to motion in its pre-generated path although it may stop, retreat or change velocity to allow coordination with other robots

Decoupled Planning with Pairwise Coordination

• The paths t1 and t2 of the first two robots are coordinated in their 2-dimensional coordination space• Results in a collision-free coordinated patht1,2

Done by using planning a pathbetween (0,0) and (1,1)

Decoupled Planning with Pairwise Coordination

• The process is repeated for paths t1,2 and t3 resulting in a coordinated path t1,2,3

• Eventually a collision-free coordinate path t1,2,…,m is generated that defines a valid coordination of all m robots

Decoupled Planning with Global Coordination

• The paths of all m robots are coordinated in an m-dimensional coordination space• Results in a collision-free path t1,2,….m

Done by planning a path from (0,0,0,…) to (1,1,1,…)

Decoupled Planning

• Advantages• Less expensive than centralized planning because lower

dimensional spaces are searched

• Disadvantages• Incomplete : Failures usually occur in the second phase as

it might not be possible to coordinate the paths generated in the first phase without collision between robots

Decoupled Planning Failure Example

• Initial and goal configurations


• Likely path generation in 1st phase


• Path coordination fails in second phase

Implemented Planners

• C-SBL – Centralized Planning• DG-SBL – Decoupled Planning with Global

Coordination• DP-SBL – Decoupled Planning with Pairwise

Coordination• Experiments conducted with groups of 2, 4 and 6

robots on 3 separate sets of initial/goal configurations

PRM Path Planner: Sampling Strategy

• SBL Planner• Single-query• Bi-directional• Lazy collision-checking

Problem I – Initial and goal configurations

Problem II – Initial and goal configurations

Problem III – Initial and goal configurations

Experimental Results

• T = average running time (seconds)• DG-SBL and DP-SBL - 20 runs per experiment• C-SBL – 100 runs per experiment

• F = number of failures• Maximum of 50,000 milestones allowed per call

to SBL


• Centralized planning had no failures• At least one failure suffered in each experiment with

decoupled planning• Failure rate increased as problems became more

complex


• pairwise coordination more unreliable than global coordination

• Failure always occurred in the 2nd stage during path coordination, a result of wrong path choices made in the 1st stage


• Similar running times for both planners in most experiments

• However, centralized planning required a lot more time than decoupled planning in 3rd problem with 6 robots

Conclusions

• Reliability – Decoupled planning can be quite unreliable particularly in tight robot coordination. Centralized planning appears to have perfect reliability.

• Planning Time – Using SBL, there is not a huge difference between the two methods

Conclusions Contd.

• Results invalidate the assumptions that loss of incompleteness with decoupled planning is fairly insignificant and can be ignored in practice.

• SBL makes usage of centralized planning for multi-robot systems practical.

• But centralized planning still requires knowledge of all robot states, which may be impossible in some settings.

Sokoban

• Objective of Robot:

To push boxes into their storage locations without getting himself or boxes stuck.

• Rules: Cannot pull, can push only one box at a time

Sokoban

Sample Sokoban Game

Documents

Motion Planning & Robot Planning