Instructor Prof. Shih-Chung Kang 2008 Spring Motion Planning (3/3) Instructor Prof. Shih-Chung Kang 2008 Spring

The motion planning problem Goal Obstacle R May 10, 2018 2

(I) Configuration Space

What is a Path? Goal obstacle obstacle obstacle 4 May 10, 2018 S.C. Kang 4 4

How to simplify this problem? What is a Path? How to simplify this problem? Use C-space May 10, 2018 S.C. Kang 5 5

Tool: Configuration Space (C-Space C) θ y goal l goal C-obstacle obstacle start start x Cartesian (real-world) space C-space θ May 10, 2018 S.C. Kang 6 6

Tool: Configuration Space(C-Space C) May 10, 2018 S.C. Kang 7 7

Configuration Space (C-space) qn q=(q1,…,qn) q3 q1 q2 May 10, 2018 S.C. Kang 8 8

Definition A robot configuration is a specification of the positions of all robot points relative to a fixed coordinate system Usually a configuration is expressed as a “vector” of position/orientation parameters May 10, 2018 S.C. Kang 9 9

Rigid Robot Example q y x workspace robot reference direction reference point x 3-parameter representation: q = (x,y,q) In a 3-D workspace q would be of the form (x,y,z,a,b,g) May 10, 2018 S.C. Kang 10 10

Articulated robot example q1 q2 q = (q1,q2,…,q10) May 10, 2018 S.C. Kang 11 11

Find the C-space of a tower crane May 10, 2018 12 12

Using C-space to simplify the motion planning of a tower crane Tower crane in Cartesian space Forward kinematics Tower crane in C-space Inverse kinematics The geometrical information to describe a tower crane Four parameters: θ1, d2, d3, and θ4 May 10, 2018 13 13

Using C-space to simplify the motion planning of a tower crane Forward kinematics Tower crane in Cartesian space Tower crane in C-space Inverse kinematics Can be represented by a point in C-space The attitude of the tower crane May 10, 2018 14 14

The benefit from using C-space (1/2) In C-space, we are able to describe the crane geometry in space using the minimal set of parameters. For example, the position in space (the attitude) of a tower crane in Cartesian space can be described by only four variables, θ1, d2, d3, and θ4. Because C-space is constructed by the four space factors, a set of the four variables is a point in C-space. A motion of a tower crane can be described by a series of these four variables, which can form a continuous line in C-space. May 10, 2018 15 15

The benefits from using C-space (2/2) After the direct and inverse kinematics of manipulators are derived, we can transfer the crane model and obstacles from the Cartesian space to a C-space. The problem of finding a collision-free erection path on a complex construction site can be simplified by finding a path that does not go into C-obstacle regions in the C-space. Because this method does not need to deal with the full geometry and kinematics information of the whole crane in the Cartesian space, the computation and complexity of the path planning problem is significantly reduced. May 10, 2018 16 16

(II) Motion planning PRM and RRTs method A tower crane example 17

Probabilistic Roadmap (PRM) local path free space milestone mb mg [Kavraki, Svetska, Latombe,Overmars, 95] May 10, 2018 S.C. Kang 18 18

Rapidly-exploring Random Trees (RRTs) free space mb mg May 10, 2018 S.C. Kang 19 19

Weaker Completeness Complete planner  Too slow Heuristic planner  Too unreliable Probabilistic completeness: If there is a solution path, the probability that the planner will find is a (fast growing) function that goes to 1 as running time increases. May 10, 2018 S.C. Kang 20 20

Issues Why random sampling? -convenient incremental scheme Smart sampling strategies -sample the points which have higher probability to be chosen in the final path. Final path smoothing May 10, 2018 S.C. Kang 21 21

How to find an erection path? The path needs to be collision-free Also needs to reachable by the crane Target Start May 10, 2018 S.C. Kang 22 22

How to find an erection path? Using Probabilistic Roadmap approach (PRM) (Latombe, 1985) Sample points within crane’s reachable area Link the points to find a collision-free path Target Start Point May 10, 2018 S.C. Kang 23 23

Demo: PRM method This video is the final project in course of Motion Planning taught by Prof. Jean-Claude Latombe. Mr. Xiaoshan Pan is another team member of the project 24

How to find an erection path? 1. QuickLink Mothod:Try linking two trees vertical to initial and goal point Tgoal target Tinit start May 10, 2018 S.C. Kang 25 25

Demo: The paths generated by QuickLink Method A collision free path Video Demo: The paths generated by QuickLink Method 26

How to find an erection path? 2. QuickGuess Mothod: Add a random middle point between two trees and try to link them by passing a random middle point Tgoal Tinit target start May 10, 2018 S.C. Kang 27 27

Demo: Collision-free paths found by using QuickGuess method A collision free path Video Demo: Collision-free paths found by using QuickGuess method 28

How to find an erection path? 3. Modified Random Method: Sample more points in the region with higher possibility to find a path Tinit Tgoal May 10, 2018 S.C. Kang 29 29

How to refine an erection path? The process to refine a collision-free path (a) (b) (c) (d) Collision-free path without any refining Eliminate redundant nodes in the path Smoothen the path reduce sharp angles Replace the straight line by curves May 10, 2018 S.C. Kang 30 30

A collision free path after refining Demo Video A collision free path after refining Demo: Refined erection paths 31

Course website http://robot.caece.net Question? Course website http://robot.caece.net