Advanced Robotics Path Planning & Navigation

Size: px
Start display at page:

Download "Advanced Robotics Path Planning & Navigation"

Transcription

1 Advanced Robotics Path Planning & Navigation 1

2 Agenda Motivation Basic Definitions Configuration Space Global Planning Local Planning Obstacle Avoidance ROS Navigation Stack 2

3 Literature Choset, Lynch, Hutchinson, Kantor, Burgard, Kavraki and Thrun. Principle of Robot Motion Theory, Algorithms and Implementation. MIT Press

4 Motivation 4

5 given a set of obstacles Piano Mover s Problem the initial position of a robot the final position of a robot goal find a path that moves the robot from the initial to final position avoids the obstacles (at all times) 5

6 Challenges in the Real World physical Laws, e.g. inertia, acceleration uncertainty, e.g. maps, observations geometric constraints, e.g. shape of a robot dynamic environment, e.g. a moving crowd 6

7 completeness Properties of Algorithms does the algorithm find a solution in finite time if one exists resolution complete probabilistic complete optimality does the algorithm find the optimal solution(s) complexity what are the computational and memory demands off-line vs. on-line pre-computing? sensor-based does the algorithm integrate a sensing step 7

8 Classical Two-Layered Architecture Path Planning low frequency map sub-goal sensor data Obstacle Avoidance Reactive Navigation motion commands high frequency 8

9 earliest and most simple sensor-based navigation biologically inspired Bug Algorithm robot is able to detect obstacles and the direction to the goal basic idea follow the straight line to the goal if hitting an obstacle circle the obstacle clockwise continue from the closest free point 9

10 Bug Algorithm 10

11 Bug Algorithm [Choset et al. 2005] 11

12 Bug Algorithm - Properties uses exhaustive search complete for finite number of hits does not need a map often suboptimal solutions Bug2 path-length: d( q start, q end ) L p i perimeter of obstacle i d( q, q ) 1. bug1 start end 5 n i 1 p i 12

13 Configuration Space to ensure that we have no collision at any time we have to represent the robot s position moreover we have to represent the position of each point of the robot we use the concept of configuration, e.g. necessary for robot arms the space of all possible configurations is the configuration space or C-space q single configuration Q entire configuration space 13

14 Example C-Space circular robot with radius r on the plane configuration can be represented as position (x,y) y x 14

15 Another Example C-Space Configuration Space Work Space 15

16 Obstacle in the Configuration Space let q be a point in the configuration space Q path planning revised: find a mapping c: [0,1] Q so that no configuration along the path intersects an obstacle a configuration space obstacle QO i is a set of configurations q at which the robot intersects WO i that is: QO i the free configuration space { q Q R( q) WO } a free path is a mapping: c: [0,1] Qf ree a semi-free path is a mapping: c: [0,1] c/(q free ) i Q Q \ ( free QO i i ) 16

17 Example Obstacle Space Ken Goldberg 17

18 Goal Building Blocks for Ground Robots Global Planner Path Trajectory Generation Intermediate Goals Local Planner Velocity Commands 18

19 given Global Planning a representation of the environment including known (static) obstacles, e.g. a map a current position (configuration) a goal position (configuration) goal find a free path from the current position (configuration) to the goal position (configuration) some assumptions holonomic motion point robot static environment no dynamic 19

20 Topological Representations represent the environment as graphs or roadmaps graph comprises nodes configurations edges (free) connections between configurations or roads either given or constructed (e.g., cell decomposition) planning = search for shortest path in graph 20

21 Geometrical Representations represent the environment by its geometry represents the workspace of the robot uses geometric primitives to represent the environment compact representation 21

22 Grid-Based Representations represent the environment as grid map grid cells are either free or occupied may represent some uncertainty represents the workspace of the robot memory demand depends on the resolution 22

23 Cell Decomposition basic Idea: convert representation into a graph and do motion planning on that graph divide the free configuration space Q free into simple connected regions named cells generate a connectivity graph with connected regions locate goal and start cells search for a path trough the graph follow the path 23

24 Trapezoid Cell Decomposition assumes polygonal environment decompose into trapezoids or triangles sort vertices in one direction iterate all vertices v i extend vertex v i up and/or down by two line segments until hitting an obstacle 24

25 Trapezoid Cell Decomposition Run Time for n vertices sorting the vertices needs O(n log(n)) time intersection done in a naïve way, we need O(n 2 ) time if we use a scan line we need only O(n log(n)) four different events and an edge list both edges of a vertex are on the left remove both from list both edges of a vertex are on the right add both to list if the upper edge is on the left and the lower edge on the right add upper edge and remove lower edge if the upper edge is on the right and the lower edge on the left add lower edge and remove upper edge perform intersection check only on edges in the list 25

26 Occupancy Grid occupancy grids are examples of an approximate cell decomposition regular grid of cells easy to implement widely used may require post planning smoothing 26

27 Occupancy Grid Alternative Representations resolution is a limiting issue too fine a grid leads to long search time too coarse a grid misses paths quad-tree adaptable cell size tree structure 27

28 A* Search most popular algorithm to find paths in graphs estimated costs function for a node n f(n) = g(n) + h(n) g(n) costs from start so far h(n) estimated costs to goal from n expand node n with lowest f(n) first two node lists: open (new), close (fully expand) if a successor node already closed ignore node if successor n is open and g (n) < g(n) then update f(n) and mark n as open continue until goal reached or no more open nodes h(n) has to underestimate the real costs 28

29 A* Search Example find the shortest path from Saabrücken to Würzburg h(n) Euclidian distance to goal 29

30 LPN Algorithm a method to find minimum cost paths based on linear programming work well on grid maps, extension of the wave-front algorithm given path as points in sample space P = {p 1, p 2,, p n } costs for traversing P navigation function represents the steepest gradient for a path P k starting at point k goal F( P) N a path P with minimum costs k k i 1 I( p i ) k 1 i 1 min F( P ) P k k A( p i, p i 1 ) 30

31 LPN Algorithm initially assign 0 to goal and to all other points obstacles contribute to I(p) goal point is active point expand path to neighbors p and calculate F (p) if F (p) < F(p) update F(p) and set p as active point continue for all active points until no one remains 31

32 A=10 A=10 A=10 A=10 A=10 A=10 A=10 A=10 I=0 C= LPN Example A=10 I=20 A=10 I=5 A=10 C= C= I=0 C= I=0 C=0 A=10 I=50 C= A=10 I=10 C= A=10 I=0 C= I=0 C= A=10 I=0 C= A=10 I=0 A=10 C= I=0 C= 32

33 Sampling-Based Approaches so far planner rely on explicit representations of Q free for complex and high-dimensional spaces this can be impractical basic idea: sample collision-free configuration and connect them to get a path multi-query planner: the planner once generate a general data structure by sampling which can be used for several path planning problems, e.g. probabilistic road maps single-query planner: sampling is done for every problem, e.g. rapidly-exploring random trees 33

34 Sampling-Based Approaches single-query planner multi-query planner 34

35 Rapidly-Exploring Random Trees (RRT) basic idea given a start configuration q init and a goal configuration q goal build a tree based on connected samples from Q free rooted in q init try to reach q goal what do we need configuration space Q start configuration q init and a goal configuration q goal collision detector: D: X true, false incremental simulator: given q(t) compute q(t + t) distance metric: ρ: Q Q [0, ) 35

36 Rapidly-Exploring Random Trees (RRT) build_rrt(q init ) T.init(q init ) for k=1 to K do q rand =random_config() extend(t, q rand ) return T extend(t,q) q near =nearest_config(q,t) if new_config(q,q near,q new,u new ) T.add_vertec(q new ) T.add_edge(q near, q new, u new ) if q new =q then return reached else return advanced return traspped 36

37 Rapidly-Exploring Random Trees (RRT) u new q new q rand q near q init 37

38 Rapidly-Exploring Random Trees (RRT) basically not goal-orientated apply some bias choose a new sample q rand randomly with probability 1 ε and as q goal randomly with probability ε grow two trees from q init and q goal and merge them the sampling strategy and the increment control the exploration of the configuration space, e.g. density of the tree RRT is probabilistic complete 38

39 OMPL: The Open Motion Planning Library open source sample-based motion planning library by Rice University provides a number of ready to use planner but also templates to implement new ones OMPL contains implementations of PRM, RRT, EST, SBL, KPIECE, OMPL supports various state (configuration) spaces OMPL is a stand-alone library but closely integrated in ROS OMPL is the standard library for arm motion planning in ROS 39

40 OMPL.app interactive GUI for motion planning problems to be solved by OMPL 40

41 Potential Field models the robot as a charged particle in a potential field U q = U att q + U rep (q), q = (x, y) the goal is an attractive field, U att (q) obstacles are a repulsive field, U rep (q) gradient of the potential field provides a force to wards towards the goal: F q = U q = U x U y 41

42 Potential Field - Properties robot rolls down a surface towards the goal easy to implement hard to find proper parameter robot may become stuck in local minima needs a map 42

43 Potential Field Example map potential path [Siegwart, Nourbakhsh, Scaramuzza, 2011, MIT Press] 43

44 performs local planning Obstacle Avoidance in general the environment is not fully modeled the environment changes dynamically obstacle avoidance may modify or replan a globally generated plan move the robot towards a goal without a global plan (e.g., Bug algorithm) it relies on information about the goal or sub-goal on the global path local context information (localization) recent sensor information (local map and obstacles) 44

45 basic idea: Vector Field Histogram generate obstacle forces which increase near obstacles generate a discrete polar histogram with n segments of that forces use a threshold to get a binary diagram threshold represents a maximum distance to include obstacles check for a valley the robot can pass, e.g. is it wide enough select the best solution according a cost function distinguish two cases (wide, narrow) determine heading angle and speed based on a candidate valley 45

46 Vector Field Histogram - Calculation for each distance senor reading d i calculate the obstacle force: m i = c i 2 a bd i c i is a uncertainty factor a, b positive constants if several sensor readings fall into one polar segment k h k = m i Ri S k binarize the histogram using a threshold h max : h k = 1 h k h max 0 otherwise 46

47 Vector Field Histogram - Selection determine candidate valley: valley:=set of successive sectors k with h k = 0 select the valley closest to the segment containing the target select the heading case 1: target (k target ) is in candidate valley, θ sol = θ target, θ_k orientation of segment k case 2: target is not in candidate valley, but number of segments is larger than a number m, k sol = k i ± m 2, k i is the sector in the valley closed to k target case 3: target is not in candidate valley, number of segments is lower than a number m, k sol = k i+k j 2, k i, k j extreme segments of valley select the velocity select v sol inversely proportional to the distance to closest obstacle 47

48 Vector Field Histogram Example situation histogram [Sicillano, Khatib, 2008, Springer] 48

49 Reactive Navigation Differential Drive t i+1 t i previous approaches did not respect the robot s kinematics and dynamics 49

50 Dynamic Window Approach the approach assumes a differential drive robot moving along arcs considers aspects of the dynamics operates in the velocity space (v, ω), V s velocity space the robot allows (maximal velocities) the approach uses the actual velocity of the robot a dynamic window of reachable velocities in the next cycle given by the dynamic of the robot calculate admissible velocities allow to stop before hitting an obstacle maximize an objective function 50

51 Admissible Velocities velocities are admissible if the robot is abele to stop before reaching an obstacle velocity space environment V a = v, ω v 2dist v, ω a t ω 2dist v, ω a r dist v, ω distance to closest obstacle a t maximal translational acceleration a r maximal rotational acceleration [Fox et al. 1997] 51

52 Reachable Velocities velocities reachable by acceleration within next time inervall (dynamic window) V d = v, ω v v a t Δt, v + a t Δt ω ω a r Δt, ω + a r Δt v, ω... actual velocity a t maximal translational acceleration a r maximal rotational acceleration [Fox et al. 1997] 52

53 DWA Search Space V s all possible velocities of the robot V a possible velocities where robot is able to stop before an obstacle V d velocities reachable the robot due to dynamics V r candidate velocities, V r = V s V a V d [Fox et al. 1997] 53

54 Selection of Velocities selection follows a heuristic drive fast and in the right direction velocities are selected to maximize an objective function: argmax α heading v, ω + β distance v, ω + γ velocity v, ω v,ω heading v, ω difference to target direction, π Δθ, difference Δθ is calculated at the pose reached after applying the velocity for Δt distance v, ω distance to the closest obstacle intersecting the arc velocity v, ω simply the translational velocity v 54

55 Global Dynamic Window Approach incorporate some features of global planning add a minimum-free function NF1 (wave-propagation) use a occupancy grid around robot update occupancy grid by range measurements extend the objective function of the DWA to stick at the greatest gradient of the navigation function avoid local minima 55

56 Combining Local and Global Methods neither global nor local methods are enough global methods fail to address local variations, uncertainty, dynamics, local methods get trapped in local minima solution: combine both methods several methods come in a global X version such as global DWA 56

57 (Rough) Frequency Range large span in frequencies in navigation Path Planning (< 0.1 Hz) Path Deformation (~ 5 Hz) Obstacle Avoidance (~ 10 Hz) Actuator Control (~ 150 Hz) system architecture have to support this variety 57

58 ROS Navigation Stack ROS provides a full navigation stack global planning A* and LPN local planning Dynamic Window Approach or Trajectory Rollout supports differential drive and omni-directional robots works with and without global localization and map can work with 2d and/or 3d maps and sensors for usage and parameter look to the tutorial 58

59 ROS Navigation Stack Overview 59

60 Some Challenges the navigation stack is trimmed towards challenging indoor office-like environments narrow passages or doors difficult 3d structures like tables or wider bases rich variety of objects in the environment unexplored areas like blind corners 60

61 Cost Maps cost maps are used to generate a path or do achieve obstacle avoidance cost maps are 2d global cost map initialized by the global map updated by sensor information map coordinate frame local cost map rolling window centered around the robot cares about local obstacles updated by sensor information odometry coordinate frame 61

62 2d versus 3d the navigation stack is able to work with laser scans (2d) and point clouds (3d) laser scans are filtered for larger geometric objects like lines remaining points are classified as obstacle ego-hits are removed point clouds are converted into a 3d occupancy grid unknown or occluded areas are treaded as forbidden areas for navigation the 3d occupancy grid is projected to a 2d one 62

63 Thank you! 63

Advanced Robotics Path Planning & Navigation

Advanced Robotics Path Planning & Navigation Advanced Robotics Path Planning & Navigation 1 Agenda Motivation Basic Definitions Configuration Space Global Planning Local Planning Obstacle Avoidance ROS Navigation Stack 2 Literature Choset, Lynch,

More information

RoboCup Rescue Summer School Navigation Tutorial

RoboCup Rescue Summer School Navigation Tutorial RoboCup Rescue Summer School 2012 Institute for Software Technology, Graz University of Technology, Austria 1 Literature Choset, Lynch, Hutchinson, Kantor, Burgard, Kavraki and Thrun. Principle of Robot

More information

Robot Motion Planning

Robot Motion Planning Robot Motion Planning slides by Jan Faigl Department of Computer Science and Engineering Faculty of Electrical Engineering, Czech Technical University in Prague lecture A4M36PAH - Planning and Games Dpt.

More information

Robot Motion Control Matteo Matteucci

Robot Motion Control Matteo Matteucci Robot Motion Control Open loop control A mobile robot is meant to move from one place to another Pre-compute a smooth trajectory based on motion segments (e.g., line and circle segments) from start to

More information

Path Planning. Marcello Restelli. Dipartimento di Elettronica e Informazione Politecnico di Milano tel:

Path Planning. Marcello Restelli. Dipartimento di Elettronica e Informazione Politecnico di Milano   tel: Marcello Restelli Dipartimento di Elettronica e Informazione Politecnico di Milano email: restelli@elet.polimi.it tel: 02 2399 3470 Path Planning Robotica for Computer Engineering students A.A. 2006/2007

More information

Autonomous Mobile Robots, Chapter 6 Planning and Navigation Where am I going? How do I get there? Localization. Cognition. Real World Environment

Autonomous Mobile Robots, Chapter 6 Planning and Navigation Where am I going? How do I get there? Localization. Cognition. Real World Environment Planning and Navigation Where am I going? How do I get there?? Localization "Position" Global Map Cognition Environment Model Local Map Perception Real World Environment Path Motion Control Competencies

More information

Motion Planning, Part III Graph Search, Part I. Howie Choset

Motion Planning, Part III Graph Search, Part I. Howie Choset Motion Planning, Part III Graph Search, Part I Howie Choset Happy President s Day The Configuration Space What it is A set of reachable areas constructed from knowledge of both the robot and the world

More information

Cognitive Robotics Robot Motion Planning Matteo Matteucci

Cognitive Robotics Robot Motion Planning Matteo Matteucci Cognitive Robotics Robot Motion Planning Robot Motion Planning eminently necessary since, by definition, a robot accomplishes tasks by moving in the real world. J.-C. Latombe (1991) Robot Motion Planning

More information

Introduction to Mobile Robotics Path Planning and Collision Avoidance. Wolfram Burgard, Maren Bennewitz, Diego Tipaldi, Luciano Spinello

Introduction to Mobile Robotics Path Planning and Collision Avoidance. Wolfram Burgard, Maren Bennewitz, Diego Tipaldi, Luciano Spinello Introduction to Mobile Robotics Path Planning and Collision Avoidance Wolfram Burgard, Maren Bennewitz, Diego Tipaldi, Luciano Spinello 1 Motion Planning Latombe (1991): is eminently necessary since, by

More information

Introduction to Mobile Robotics Path Planning and Collision Avoidance

Introduction to Mobile Robotics Path Planning and Collision Avoidance Introduction to Mobile Robotics Path Planning and Collision Avoidance Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Giorgio Grisetti, Kai Arras 1 Motion Planning Latombe (1991): eminently necessary

More information

Motion Planning. Howie CHoset

Motion Planning. Howie CHoset Motion Planning Howie CHoset Questions Where are we? Where do we go? Which is more important? Encoders Encoders Incremental Photodetector Encoder disk LED Photoemitter Encoders - Incremental Encoders -

More information

Visual Navigation for Flying Robots. Motion Planning

Visual Navigation for Flying Robots. Motion Planning Computer Vision Group Prof. Daniel Cremers Visual Navigation for Flying Robots Motion Planning Dr. Jürgen Sturm Motivation: Flying Through Forests 3 1 2 Visual Navigation for Flying Robots 2 Motion Planning

More information

Introduction to State-of-the-art Motion Planning Algorithms. Presented by Konstantinos Tsianos

Introduction to State-of-the-art Motion Planning Algorithms. Presented by Konstantinos Tsianos Introduction to State-of-the-art Motion Planning Algorithms Presented by Konstantinos Tsianos Robots need to move! Motion Robot motion must be continuous Geometric constraints Dynamic constraints Safety

More information

Introduction to Information Science and Technology (IST) Part IV: Intelligent Machines and Robotics Planning

Introduction to Information Science and Technology (IST) Part IV: Intelligent Machines and Robotics Planning Introduction to Information Science and Technology (IST) Part IV: Intelligent Machines and Robotics Planning Sören Schwertfeger / 师泽仁 ShanghaiTech University ShanghaiTech University - SIST - 10.05.2017

More information

Introduction to Mobile Robotics Path and Motion Planning. Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello

Introduction to Mobile Robotics Path and Motion Planning. Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello Introduction to Mobile Robotics Path and Motion Planning Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello 1 Motion Planning Latombe (1991): eminently necessary since,

More information

Algorithms for Sensor-Based Robotics: Sampling-Based Motion Planning

Algorithms for Sensor-Based Robotics: Sampling-Based Motion Planning Algorithms for Sensor-Based Robotics: Sampling-Based Motion Planning Computer Science 336 http://www.cs.jhu.edu/~hager/teaching/cs336 Professor Hager http://www.cs.jhu.edu/~hager Recall Earlier Methods

More information

Robot Motion Planning

Robot Motion Planning Robot Motion Planning James Bruce Computer Science Department Carnegie Mellon University April 7, 2004 Agent Planning An agent is a situated entity which can choose and execute actions within in an environment.

More information

Visibility Graph. How does a Mobile Robot get from A to B?

Visibility Graph. How does a Mobile Robot get from A to B? Robot Path Planning Things to Consider: Spatial reasoning/understanding: robots can have many dimensions in space, obstacles can be complicated Global Planning: Do we know the environment apriori? Online

More information

Sung-Eui Yoon ( 윤성의 )

Sung-Eui Yoon ( 윤성의 ) Path Planning for Point Robots Sung-Eui Yoon ( 윤성의 ) Course URL: http://sglab.kaist.ac.kr/~sungeui/mpa Class Objectives Motion planning framework Classic motion planning approaches 2 3 Configuration Space:

More information

Path Planning for Point Robots. NUS CS 5247 David Hsu

Path Planning for Point Robots. NUS CS 5247 David Hsu Path Planning for Point Robots NUS CS 5247 David Hsu Problem Input Robot represented as a point in the plane Obstacles represented as polygons Initial and goal positions Output A collision-free path between

More information

Algorithms for Sensor-Based Robotics: Sampling-Based Motion Planning

Algorithms for Sensor-Based Robotics: Sampling-Based Motion Planning Algorithms for Sensor-Based Robotics: Sampling-Based Motion Planning Computer Science 336 http://www.cs.jhu.edu/~hager/teaching/cs336 Professor Hager http://www.cs.jhu.edu/~hager Recall Earlier Methods

More information

ECE276B: Planning & Learning in Robotics Lecture 5: Configuration Space

ECE276B: Planning & Learning in Robotics Lecture 5: Configuration Space ECE276B: Planning & Learning in Robotics Lecture 5: Configuration Space Lecturer: Nikolay Atanasov: natanasov@ucsd.edu Teaching Assistants: Tianyu Wang: tiw161@eng.ucsd.edu Yongxi Lu: yol070@eng.ucsd.edu

More information

Probabilistic Methods for Kinodynamic Path Planning

Probabilistic Methods for Kinodynamic Path Planning 16.412/6.834J Cognitive Robotics February 7 th, 2005 Probabilistic Methods for Kinodynamic Path Planning Based on Past Student Lectures by: Paul Elliott, Aisha Walcott, Nathan Ickes and Stanislav Funiak

More information

Sampling-based Planning 2

Sampling-based Planning 2 RBE MOTION PLANNING Sampling-based Planning 2 Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering http://users.wpi.edu/~zli11 Problem with KD-tree RBE MOTION PLANNING Curse of dimension

More information

Configuration Space of a Robot

Configuration Space of a Robot Robot Path Planning Overview: 1. Visibility Graphs 2. Voronoi Graphs 3. Potential Fields 4. Sampling-Based Planners PRM: Probabilistic Roadmap Methods RRTs: Rapidly-exploring Random Trees Configuration

More information

Humanoid Robotics. Inverse Kinematics and Whole-Body Motion Planning. Maren Bennewitz

Humanoid Robotics. Inverse Kinematics and Whole-Body Motion Planning. Maren Bennewitz Humanoid Robotics Inverse Kinematics and Whole-Body Motion Planning Maren Bennewitz 1 Motivation Planning for object manipulation Whole-body motion to reach a desired goal configuration Generate a sequence

More information

Geometric Path Planning McGill COMP 765 Oct 12 th, 2017

Geometric Path Planning McGill COMP 765 Oct 12 th, 2017 Geometric Path Planning McGill COMP 765 Oct 12 th, 2017 The Motion Planning Problem Intuition: Find a safe path/trajectory from start to goal More precisely: A path is a series of robot configurations

More information

Path Planning. Jacky Baltes Dept. of Computer Science University of Manitoba 11/21/10

Path Planning. Jacky Baltes Dept. of Computer Science University of Manitoba   11/21/10 Path Planning Jacky Baltes Autonomous Agents Lab Department of Computer Science University of Manitoba Email: jacky@cs.umanitoba.ca http://www.cs.umanitoba.ca/~jacky Path Planning Jacky Baltes Dept. of

More information

Navigation and Metric Path Planning

Navigation and Metric Path Planning Navigation and Metric Path Planning October 4, 2011 Minerva tour guide robot (CMU): Gave tours in Smithsonian s National Museum of History Example of Minerva s occupancy map used for navigation Objectives

More information

Spring 2010: Lecture 9. Ashutosh Saxena. Ashutosh Saxena

Spring 2010: Lecture 9. Ashutosh Saxena. Ashutosh Saxena CS 4758/6758: Robot Learning Spring 2010: Lecture 9 Why planning and control? Video Typical Architecture Planning 0.1 Hz Control 50 Hz Does it apply to all robots and all scenarios? Previous Lecture: Potential

More information

Non-holonomic Planning

Non-holonomic Planning Non-holonomic Planning Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering http://users.wpi.edu/~zli11 Recap We have learned about RRTs. q new q init q near q rand But the standard

More information

Sampling-Based Robot Motion Planning. Lydia Kavraki Department of Computer Science Rice University Houston, TX USA

Sampling-Based Robot Motion Planning. Lydia Kavraki Department of Computer Science Rice University Houston, TX USA Sampling-Based Robot Motion Planning Lydia Kavraki Department of Computer Science Rice University Houston, TX USA Motion planning: classical setting Go from Start to Goal without collisions and while respecting

More information

Lecture 3: Motion Planning (cont.)

Lecture 3: Motion Planning (cont.) CS 294-115 Algorithmic Human-Robot Interaction Fall 2016 Lecture 3: Motion Planning (cont.) Scribes: Molly Nicholas, Chelsea Zhang 3.1 Previously in class... Recall that we defined configuration spaces.

More information

Motion Planning, Part IV Graph Search Part II. Howie Choset

Motion Planning, Part IV Graph Search Part II. Howie Choset Motion Planning, Part IV Graph Search Part II Howie Choset Map-Based Approaches: Properties of a roadmap: Roadmap Theory Accessibility: there exists a collision-free path from the start to the road map

More information

Workspace-Guided Rapidly-Exploring Random Tree Method for a Robot Arm

Workspace-Guided Rapidly-Exploring Random Tree Method for a Robot Arm WorkspaceGuided RapidlyExploring Random Tree Method for a Robot Arm Jaesik Choi choi31@cs.uiuc.edu August 12, 2007 Abstract Motion planning for robotic arms is important for real, physical world applications.

More information

Trajectory Optimization

Trajectory Optimization Trajectory Optimization Jane Li Assistant Professor Mechanical Engineering & Robotics Engineering http://users.wpi.edu/~zli11 Recap We heard about RRT*, a sampling-based planning in high-dimensional cost

More information

Final Exam Practice Fall Semester, 2012

Final Exam Practice Fall Semester, 2012 COS 495 - Autonomous Robot Navigation Final Exam Practice Fall Semester, 2012 Duration: Total Marks: 70 Closed Book 2 hours Start Time: End Time: By signing this exam, I agree to the honor code Name: Signature:

More information

Lecture 3: Motion Planning 2

Lecture 3: Motion Planning 2 CS 294-115 Algorithmic Human-Robot Interaction Fall 2017 Lecture 3: Motion Planning 2 Scribes: David Chan and Liting Sun - Adapted from F2016 Notes by Molly Nicholas and Chelsea Zhang 3.1 Previously Recall

More information

Navigation methods and systems

Navigation methods and systems Navigation methods and systems Navigare necesse est Content: Navigation of mobile robots a short overview Maps Motion Planning SLAM (Simultaneous Localization and Mapping) Navigation of mobile robots a

More information

Jane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute

Jane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute Jane Li Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute (3 pts) How to generate Delaunay Triangulation? (3 pts) Explain the difference

More information

6.141: Robotics systems and science Lecture 10: Implementing Motion Planning

6.141: Robotics systems and science Lecture 10: Implementing Motion Planning 6.141: Robotics systems and science Lecture 10: Implementing Motion Planning Lecture Notes Prepared by N. Roy and D. Rus EECS/MIT Spring 2011 Reading: Chapter 3, and Craig: Robotics http://courses.csail.mit.edu/6.141/!

More information

Principles of Robot Motion

Principles of Robot Motion Principles of Robot Motion Theory, Algorithms, and Implementation Howie Choset, Kevin Lynch, Seth Hutchinson, George Kantor, Wolfram Burgard, Lydia Kavraki, and Sebastian Thrun A Bradford Book The MIT

More information

Spring Localization II. Roland Siegwart, Margarita Chli, Juan Nieto, Nick Lawrance. ASL Autonomous Systems Lab. Autonomous Mobile Robots

Spring Localization II. Roland Siegwart, Margarita Chli, Juan Nieto, Nick Lawrance. ASL Autonomous Systems Lab. Autonomous Mobile Robots Spring 2018 Localization II Localization I 16.04.2018 1 knowledge, data base mission commands Localization Map Building environment model local map position global map Cognition Path Planning path Perception

More information

Obstacle Avoidance (Local Path Planning)

Obstacle Avoidance (Local Path Planning) Obstacle Avoidance (Local Path Planning) The goal of the obstacle avoidance algorithms is to avoid collisions with obstacles It is usually based on local map Often implemented as a more or less independent

More information

Motion Planning for Humanoid Robots

Motion Planning for Humanoid Robots Motion Planning for Humanoid Robots Presented by: Li Yunzhen What is Humanoid Robots & its balance constraints? Human-like Robots A configuration q is statically-stable if the projection of mass center

More information

Computer Game Programming Basic Path Finding

Computer Game Programming Basic Path Finding 15-466 Computer Game Programming Basic Path Finding Robotics Institute Path Planning Sven Path Planning needs to be very fast (especially for games with many characters) needs to generate believable paths

More information

Introduction to Robotics

Introduction to Robotics Jianwei Zhang zhang@informatik.uni-hamburg.de Universität Hamburg Fakultät für Mathematik, Informatik und Naturwissenschaften Technische Aspekte Multimodaler Systeme 05. July 2013 J. Zhang 1 Task-level

More information

Introduction to Robotics

Introduction to Robotics Introduction to Robotics Ph.D. Antonio Marin-Hernandez Artificial Intelligence Department Universidad Veracruzana Sebastian Camacho # 5 Xalapa, Veracruz Robotics Action and Perception LAAS-CNRS 7, av du

More information

Human-Oriented Robotics. Robot Motion Planning. Kai Arras Social Robotics Lab, University of Freiburg

Human-Oriented Robotics. Robot Motion Planning. Kai Arras Social Robotics Lab, University of Freiburg Robot Motion Planning Kai Arras, University of Freiburg 1 Contents Introduction Configuration space Combinatorial planning Sampling-based planning Potential fields methods A, Any-Angle A, D/D Lite Dynamic

More information

Spring Localization II. Roland Siegwart, Margarita Chli, Martin Rufli. ASL Autonomous Systems Lab. Autonomous Mobile Robots

Spring Localization II. Roland Siegwart, Margarita Chli, Martin Rufli. ASL Autonomous Systems Lab. Autonomous Mobile Robots Spring 2016 Localization II Localization I 25.04.2016 1 knowledge, data base mission commands Localization Map Building environment model local map position global map Cognition Path Planning path Perception

More information

Humanoid Robotics. Inverse Kinematics and Whole-Body Motion Planning. Maren Bennewitz

Humanoid Robotics. Inverse Kinematics and Whole-Body Motion Planning. Maren Bennewitz Humanoid Robotics Inverse Kinematics and Whole-Body Motion Planning Maren Bennewitz 1 Motivation Plan a sequence of configurations (vector of joint angle values) that let the robot move from its current

More information

Motion Planning. COMP3431 Robot Software Architectures

Motion Planning. COMP3431 Robot Software Architectures Motion Planning COMP3431 Robot Software Architectures Motion Planning Task Planner can tell the robot discrete steps but can t say how to execute them Discrete actions must be turned into operations in

More information

6.141: Robotics systems and science Lecture 9: Configuration Space and Motion Planning

6.141: Robotics systems and science Lecture 9: Configuration Space and Motion Planning 6.141: Robotics systems and science Lecture 9: Configuration Space and Motion Planning Lecture Notes Prepared by Daniela Rus EECS/MIT Spring 2012 Figures by Nancy Amato, Rodney Brooks, Vijay Kumar Reading:

More information

Motion Planning. Howie CHoset

Motion Planning. Howie CHoset Motion Planning Howie CHoset What is Motion Planning? What is Motion Planning? Determining where to go Overview The Basics Motion Planning Statement The World and Robot Configuration Space Metrics Algorithms

More information

Planning: Part 1 Classical Planning

Planning: Part 1 Classical Planning Planning: Part 1 Classical Planning Computer Science 6912 Department of Computer Science Memorial University of Newfoundland July 12, 2016 COMP 6912 (MUN) Planning July 12, 2016 1 / 9 Planning Localization

More information

6.141: Robotics systems and science Lecture 10: Motion Planning III

6.141: Robotics systems and science Lecture 10: Motion Planning III 6.141: Robotics systems and science Lecture 10: Motion Planning III Lecture Notes Prepared by N. Roy and D. Rus EECS/MIT Spring 2012 Reading: Chapter 3, and Craig: Robotics http://courses.csail.mit.edu/6.141/!

More information

Mobile Robots Locomotion

Mobile Robots Locomotion Mobile Robots Locomotion Institute for Software Technology 1 Course Outline 1. Introduction to Mobile Robots 2. Locomotion 3. Sensors 4. Localization 5. Environment Modelling 6. Reactive Navigation 2 Today

More information

Can work in a group of at most 3 students.! Can work individually! If you work in a group of 2 or 3 students,!

Can work in a group of at most 3 students.! Can work individually! If you work in a group of 2 or 3 students,! Assignment 1 is out! Due: 26 Aug 23:59! Submit in turnitin! Code + report! Can work in a group of at most 3 students.! Can work individually! If you work in a group of 2 or 3 students,! Each member must

More information

Exam in DD2426 Robotics and Autonomous Systems

Exam in DD2426 Robotics and Autonomous Systems Exam in DD2426 Robotics and Autonomous Systems Lecturer: Patric Jensfelt KTH, March 16, 2010, 9-12 No aids are allowed on the exam, i.e. no notes, no books, no calculators, etc. You need a minimum of 20

More information

Autonomous and Mobile Robotics Prof. Giuseppe Oriolo. Motion Planning 1 Retraction and Cell Decomposition

Autonomous and Mobile Robotics Prof. Giuseppe Oriolo. Motion Planning 1 Retraction and Cell Decomposition Autonomous and Mobile Robotics Prof. Giuseppe Oriolo Motion Planning 1 Retraction and Cell Decomposition motivation robots are expected to perform tasks in workspaces populated by obstacles autonomy requires

More information

6.141: Robotics systems and science Lecture 9: Configuration Space and Motion Planning

6.141: Robotics systems and science Lecture 9: Configuration Space and Motion Planning 6.141: Robotics systems and science Lecture 9: Configuration Space and Motion Planning Lecture Notes Prepared by Daniela Rus EECS/MIT Spring 2011 Figures by Nancy Amato, Rodney Brooks, Vijay Kumar Reading:

More information

EE631 Cooperating Autonomous Mobile Robots

EE631 Cooperating Autonomous Mobile Robots EE631 Cooperating Autonomous Mobile Robots Lecture 3: Path Planning Algorithm Prof. Yi Guo ECE Dept. Plan Representing the Space Path Planning Methods A* Search Algorithm D* Search Algorithm Representing

More information

CMU-Q Lecture 4: Path Planning. Teacher: Gianni A. Di Caro

CMU-Q Lecture 4: Path Planning. Teacher: Gianni A. Di Caro CMU-Q 15-381 Lecture 4: Path Planning Teacher: Gianni A. Di Caro APPLICATION: MOTION PLANNING Path planning: computing a continuous sequence ( a path ) of configurations (states) between an initial configuration

More information

Uncertainties: Representation and Propagation & Line Extraction from Range data

Uncertainties: Representation and Propagation & Line Extraction from Range data 41 Uncertainties: Representation and Propagation & Line Extraction from Range data 42 Uncertainty Representation Section 4.1.3 of the book Sensing in the real world is always uncertain How can uncertainty

More information

Humanoid Robotics. Path Planning and Walking. Maren Bennewitz

Humanoid Robotics. Path Planning and Walking. Maren Bennewitz Humanoid Robotics Path Planning and Walking Maren Bennewitz 1 Introduction Given the robot s pose in a model of the environment Compute a path to a target location First: 2D path in a 2D grid map representation

More information

Unit 5: Part 1 Planning

Unit 5: Part 1 Planning Unit 5: Part 1 Planning Computer Science 4766/6778 Department of Computer Science Memorial University of Newfoundland March 25, 2014 COMP 4766/6778 (MUN) Planning March 25, 2014 1 / 9 Planning Localization

More information

CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University. Course Logistics. Instructor: Jingjin Yu

CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University. Course Logistics. Instructor: Jingjin Yu CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University Course Logistics Instructor: Jingjin Yu Logistics, etc. General Lectures: Noon-1:20pm Tuesdays and Fridays, SEC 118 Instructor:

More information

MOBILE ROBOTS NAVIGATION, MAPPING & LOCALIZATION: PART I

MOBILE ROBOTS NAVIGATION, MAPPING & LOCALIZATION: PART I MOBILE ROBOTS NAVIGATION, MAPPING & LOCALIZATION: PART I Lee Gim Hee* CML DSO National Laboratories 20 Science Park Drive Singapore 118230 Tel: (+65) 6796 8427 Email: lgimhee@dso.org.sg Marcelo H. Ang

More information

Lecture Schedule Week Date Lecture (W: 3:05p-4:50, 7-222)

Lecture Schedule Week Date Lecture (W: 3:05p-4:50, 7-222) 2017 School of Information Technology and Electrical Engineering at the University of Queensland Lecture Schedule Week Date Lecture (W: 3:05p-4:50, 7-222) 1 26-Jul Introduction + 2 2-Aug Representing Position

More information

Today MAPS AND MAPPING. Features. process of creating maps. More likely features are things that can be extracted from images:

Today MAPS AND MAPPING. Features. process of creating maps. More likely features are things that can be extracted from images: MAPS AND MAPPING Features In the first class we said that navigation begins with what the robot can see. There are several forms this might take, but it will depend on: What sensors the robot has What

More information

L17. OCCUPANCY MAPS. NA568 Mobile Robotics: Methods & Algorithms

L17. OCCUPANCY MAPS. NA568 Mobile Robotics: Methods & Algorithms L17. OCCUPANCY MAPS NA568 Mobile Robotics: Methods & Algorithms Today s Topic Why Occupancy Maps? Bayes Binary Filters Log-odds Occupancy Maps Inverse sensor model Learning inverse sensor model ML map

More information

AUTONOMOUS SYSTEMS. LOCALIZATION, MAPPING & SIMULTANEOUS LOCALIZATION AND MAPPING Part V Mapping & Occupancy Grid Mapping

AUTONOMOUS SYSTEMS. LOCALIZATION, MAPPING & SIMULTANEOUS LOCALIZATION AND MAPPING Part V Mapping & Occupancy Grid Mapping AUTONOMOUS SYSTEMS LOCALIZATION, MAPPING & SIMULTANEOUS LOCALIZATION AND MAPPING Part V Mapping & Occupancy Grid Mapping Maria Isabel Ribeiro Pedro Lima with revisions introduced by Rodrigo Ventura Instituto

More information

Scan Matching. Pieter Abbeel UC Berkeley EECS. Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics

Scan Matching. Pieter Abbeel UC Berkeley EECS. Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics Scan Matching Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics Scan Matching Overview Problem statement: Given a scan and a map, or a scan and a scan,

More information

Motion Planning with Dynamics, Physics based Simulations, and Linear Temporal Objectives. Erion Plaku

Motion Planning with Dynamics, Physics based Simulations, and Linear Temporal Objectives. Erion Plaku Motion Planning with Dynamics, Physics based Simulations, and Linear Temporal Objectives Erion Plaku Laboratory for Computational Sensing and Robotics Johns Hopkins University Frontiers of Planning The

More information

Autonomous Robotics 6905

Autonomous Robotics 6905 6905 Lecture 5: to Path-Planning and Navigation Dalhousie University i October 7, 2011 1 Lecture Outline based on diagrams and lecture notes adapted from: Probabilistic bili i Robotics (Thrun, et. al.)

More information

Introduction to Intelligent System ( , Fall 2017) Instruction for Assignment 2 for Term Project. Rapidly-exploring Random Tree and Path Planning

Introduction to Intelligent System ( , Fall 2017) Instruction for Assignment 2 for Term Project. Rapidly-exploring Random Tree and Path Planning Instruction for Assignment 2 for Term Project Rapidly-exploring Random Tree and Path Planning Introduction The objective of this semester s term project is to implement a path planning algorithm for a

More information

Spring 2016 :: :: Robot Autonomy :: Team 7 Motion Planning for Autonomous All-Terrain Vehicle

Spring 2016 :: :: Robot Autonomy :: Team 7 Motion Planning for Autonomous All-Terrain Vehicle Spring 2016 :: 16662 :: Robot Autonomy :: Team 7 Motion Planning for Autonomous All-Terrain Vehicle Guan-Horng Liu, Samuel Wang, Shu-Kai Lin, Chris Wang, Tiffany May Advisor : Mr. George Kantor OUTLINE

More information

Path Planning. Ioannis Rekleitis

Path Planning. Ioannis Rekleitis Path Planning Ioannis Rekleitis Outline Path Planning Visibility Graph Potential Fields Bug Algorithms Skeletons/Voronoi Graphs C-Space CSCE-574 Robotics 2 Mo+on Planning The ability to go from A to B

More information

Approximate path planning. Computational Geometry csci3250 Laura Toma Bowdoin College

Approximate path planning. Computational Geometry csci3250 Laura Toma Bowdoin College Approximate path planning Computational Geometry csci3250 Laura Toma Bowdoin College Outline Path planning Combinatorial Approximate Combinatorial path planning Idea: Compute free C-space combinatorially

More information

CS Path Planning

CS Path Planning Why Path Planning? CS 603 - Path Planning Roderic A. Grupen 4/13/15 Robotics 1 4/13/15 Robotics 2 Why Motion Planning? Origins of Motion Planning Virtual Prototyping! Character Animation! Structural Molecular

More information

Road Map Methods. Including material from Howie Choset / G.D. Hager S. Leonard

Road Map Methods. Including material from Howie Choset / G.D. Hager S. Leonard Road Map Methods Including material from Howie Choset The Basic Idea Capture the connectivity of Q free by a graph or network of paths. 16-735, Howie Choset, with significant copying from who loosely based

More information

Robot Motion Planning and (a little) Computational Geometry

Robot Motion Planning and (a little) Computational Geometry Images taken from slides b B. Baazit, G. Dudek, J. C. Latombe and A. Moore Robot Motion Planning and (a little) Computational Geometr Topics: Transforms Topological Methods Configuration space Skeletonization

More information

Impact of Workspace Decompositions on Discrete Search Leading Continuous Exploration (DSLX) Motion Planning

Impact of Workspace Decompositions on Discrete Search Leading Continuous Exploration (DSLX) Motion Planning 2008 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, May 19-23, 2008 Impact of Workspace Decompositions on Discrete Search Leading Continuous Exploration (DSLX) Motion Planning

More information

Roadmaps. Vertex Visibility Graph. Reduced visibility graph, i.e., not including segments that extend into obstacles on either side.

Roadmaps. Vertex Visibility Graph. Reduced visibility graph, i.e., not including segments that extend into obstacles on either side. Roadmaps Vertex Visibility Graph Full visibility graph Reduced visibility graph, i.e., not including segments that extend into obstacles on either side. (but keeping endpoints roads) what else might we

More information

Robotics Tasks. CS 188: Artificial Intelligence Spring Manipulator Robots. Mobile Robots. Degrees of Freedom. Sensors and Effectors

Robotics Tasks. CS 188: Artificial Intelligence Spring Manipulator Robots. Mobile Robots. Degrees of Freedom. Sensors and Effectors CS 188: Artificial Intelligence Spring 2006 Lecture 5: Robot Motion Planning 1/31/2006 Dan Klein UC Berkeley Many slides from either Stuart Russell or Andrew Moore Motion planning (today) How to move from

More information

COS Lecture 13 Autonomous Robot Navigation

COS Lecture 13 Autonomous Robot Navigation COS 495 - Lecture 13 Autonomous Robot Navigation Instructor: Chris Clark Semester: Fall 2011 1 Figures courtesy of Siegwart & Nourbakhsh Control Structure Prior Knowledge Operator Commands Localization

More information

Simultaneous Localization and Mapping (SLAM)

Simultaneous Localization and Mapping (SLAM) Simultaneous Localization and Mapping (SLAM) RSS Lecture 16 April 8, 2013 Prof. Teller Text: Siegwart and Nourbakhsh S. 5.8 SLAM Problem Statement Inputs: No external coordinate reference Time series of

More information

Outline. Robotics Planning. Example: Getting to DC from Atlanta. Introduction

Outline. Robotics Planning. Example: Getting to DC from Atlanta. Introduction Outline Robotics Planning Henrik I Christensen Introduction Representations Planning Basics Search methods Evaluation (or the lack thereof) Summary Introduction Example: Getting to DC from Atlanta Generation

More information

University of Nevada, Reno. Dynamic Path Planning and Replanning for Mobile Robot Team Using RRT*

University of Nevada, Reno. Dynamic Path Planning and Replanning for Mobile Robot Team Using RRT* University of Nevada, Reno Dynamic Path Planning and Replanning for Mobile Robot Team Using RRT* A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer

More information

Beyond A*: Speeding up pathfinding through hierarchical abstraction. Daniel Harabor NICTA & The Australian National University

Beyond A*: Speeding up pathfinding through hierarchical abstraction. Daniel Harabor NICTA & The Australian National University Beyond A*: Speeding up pathfinding through hierarchical abstraction Daniel Harabor NICTA & The Australian National University Motivation Myth: Pathfinding is a solved problem. Motivation Myth: Pathfinding

More information

Mobile Robots: An Introduction.

Mobile Robots: An Introduction. Mobile Robots: An Introduction Amirkabir University of Technology Computer Engineering & Information Technology Department http://ce.aut.ac.ir/~shiry/lecture/robotics-2004/robotics04.html Introduction

More information

Local Search Methods. CS 188: Artificial Intelligence Fall Announcements. Hill Climbing. Hill Climbing Diagram. Today

Local Search Methods. CS 188: Artificial Intelligence Fall Announcements. Hill Climbing. Hill Climbing Diagram. Today CS 188: Artificial Intelligence Fall 2006 Lecture 5: Robot Motion Planning 9/14/2006 Local Search Methods Queue-based algorithms keep fallback options (backtracking) Local search: improve what you have

More information

COMPARISON OF ROBOT NAVIGATION METHODS USING PERFORMANCE METRICS

COMPARISON OF ROBOT NAVIGATION METHODS USING PERFORMANCE METRICS COMPARISON OF ROBOT NAVIGATION METHODS USING PERFORMANCE METRICS Adriano Flores Dantas, Rodrigo Porfírio da Silva Sacchi, Valguima V. V. A. Odakura Faculdade de Ciências Exatas e Tecnologia (FACET) Universidade

More information

Simultaneous Localization

Simultaneous Localization Simultaneous Localization and Mapping (SLAM) RSS Technical Lecture 16 April 9, 2012 Prof. Teller Text: Siegwart and Nourbakhsh S. 5.8 Navigation Overview Where am I? Where am I going? Localization Assumed

More information

Part I Part 1 Sampling-based Motion Planning

Part I Part 1 Sampling-based Motion Planning Overview of the Lecture Randomized Sampling-based Motion Planning Methods Jan Faigl Department of Computer Science Faculty of Electrical Engineering Czech Technical University in Prague Lecture 05 B4M36UIR

More information

ME 597/747 Autonomous Mobile Robots. Mid Term Exam. Duration: 2 hour Total Marks: 100

ME 597/747 Autonomous Mobile Robots. Mid Term Exam. Duration: 2 hour Total Marks: 100 ME 597/747 Autonomous Mobile Robots Mid Term Exam Duration: 2 hour Total Marks: 100 Instructions: Read the exam carefully before starting. Equations are at the back, but they are NOT necessarily valid

More information

Part I Part 1 Sampling-based Motion Planning

Part I Part 1 Sampling-based Motion Planning Overview of the Lecture Randomized Sampling-based Motion Planning Methods Jan Faigl Department of Computer Science Faculty of Electrical Engineering Czech Technical University in Prague Lecture 06 B4M36UIR

More information

Efficient Interpolated Path Planning of Mobile Robots based on Occupancy Grid Maps

Efficient Interpolated Path Planning of Mobile Robots based on Occupancy Grid Maps Efficient Interpolated Path Planning of Mobile Robots based on Occupancy Grid Maps Marija Ðakulović Mijo Čikeš Ivan Petrović University of Zagreb, Faculty of Electrical Engineering and Computing, Department

More information

Planning in Mobile Robotics

Planning in Mobile Robotics Planning in Mobile Robotics Part I. Miroslav Kulich Intelligent and Mobile Robotics Group Gerstner Laboratory for Intelligent Decision Making and Control Czech Technical University in Prague Tuesday 26/07/2011

More information

Jane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute

Jane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute Jane Li Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute A search-algorithm prioritizes and expands the nodes in its open list items by

More information