Motion Planning & Robot Planning Prof.: S. Shiry Mohsen gandomkar M.Sc Artificial Intelligence Department 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 The Configuration Space Configuration Space of A is the space (C) of all possible configurations of A. C Cfree qgoal Cobs qstart For a point robot moving in 2-D plane, C-space is

For a point robot moving in 3-D, the C-space is The Configuration Space C y Cfree qgoal Z Cobs qstart x For a point robot moving in 3-D, the C-space is What is the difference between Euclidean space and C-space?

The Configuration 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.

Algorithmic Approaches Complete Algorithms Probabilistic Algorithms Heuristic Algorithms

Complete Algorithms 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

Probabilistic 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

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 local path free space milestone [Kavraki, Svetska, Latombe,Overmars, 96]

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 C-obstacle Free space

PRM - Learning Phase C-obstacle Free space

PRM - Learning Phase C-obstacle Free space milestones

PRM - Learning Phase C-obstacle Free space milestones

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 & obstacles Output: roadmap G = (V, E) 1: V   and E  . 2: repeat 3: q  a configuration sampled uniformly at random from C. 4: if CLEAR(q)then 5: 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)then 8: Add an edge between q and q’ to E.

Difficulty Many small connected components

Resampling (expansion) Failure rate Weight Resampling probability

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 Find a shortest path in the roadmap Check whether the nodes and edges in the path are collision. 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 q2L qstart qgoal q2H q1L q1H Motion to goal Boundary following Shortest path to goal

Bug2 - example q2L qstart qgoal q1L q2H q1H Motion to goal Boundary following Line connecting start and goal

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 Adjusted bug algorithm

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 Adjusted bug algorithm Bug M1

Bug1 vs. Bug2 Bug1 Bug2 Exhaustive search Optimal leave point Performs better with complex obstacles Path length : n = # of obstacles Pi = perimeter of obstacle i Bug2 Opportunistic (greedy) search Performs better with simple obstacles Path length : ni = # of times the start-goal line intersects obstacle i

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.

Tangent Bug algorithm 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)

Tangent Bug algorithm Motion to goal t = 1 t = 2 t = 3 o1 o2 t = 4 o1

boundary following – define two distances: 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

Tangent Bug algorithm Boundary following Motion to goal M goal

Tangent Bug - example qgoal qstart 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 Disadvantages 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 path between (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 Disadvantages 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

Decoupled Planning Failure Example Likely path generation in 1st phase

Decoupled Planning Failure Example 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

Experimental Results Centralized planning had no failures At least one failure suffered in each experiment with decoupled planning Failure rate increased as problems became more complex

Experimental Results 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

Experimental Results 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