RRT RRT and Recent Advancements d R t Ad t Sung-Eui Yoon ( 윤성의 ) ( 윤성의 ) C Course URL: URL http://sglab.kaist.ac.kr/~sungeui/MPA
Class Objectives Class Objectives ● Understand the RRT technique and its Understand the RRT technique and its recent advancements 2
RRT-Connect: An Efficient Approach to Single-Query Path Planning Planning James Kuffner, Steven LaValle I CRA 2000 # of citation: more 600 o c tat o o e 600 Initial slides from TaeJoon Kim 3
Goal Goal ● Present an efficient randomized path Present an efficient randomized path planning algorithm for single-query problems problems ● Converges quickly ● Probabilistically complete y p ● Works well in high-dimensional C-space 4
Motivation – Performance vs Reliability Performance vs. Reliability ● Complete algorithms [Schwartz and M. Complete algorithms [Schwartz and M Sharir 83, Canny 88] ● Most reliable needs high computational power ● Most reliable, needs high computational power ● Only used to low-dimensional C-space ● Randomized potential field [Barraquand ● Randomized potential field [Barraquand and Latombe 91] ● Greedy & relaxation approach y pp ● Fast in many cases, but not in every case ● Probabilistic roadmap [Kavraki et al. 96] p [ ] ● Reliable, but needs preprocessing ● Good for multiple-query problems 5
Approach Approach ● Design a simple, reliable, and fast Design a simple reliable and fast algorithm for single-query problems ● Use RRT (Rapidly-exploring Random Trees) ● Use RRT (Rapidly-exploring Random Trees) [LaValle 98] for reliability ● Develop a greedy heuristic to converge quickly p g y g q y 6
Rapidly Exploring Random Tree Rapidly-Exploring Random Tree ● A growing tree from an initial state A growing tree from an initial state 7
RRT Construction Algorithm RRT Construction Algorithm ● Extend a new vertex in each iteration Extend a new vertex in each iteration ε q new q near q q init 8
Advantages of RRT Advantages of RRT ● Biased toward unexplored space Biased toward unexplored space ● Probabilistically complete ● Always connected ● Can handle nonholonomic constraints and hi h d high degrees of freedom f f d 9
Outline Outline ● I ntroduction I ntroduction ● Rapidly-exploring Random Tree ● Overview ● RRT-Connect Algorithm ● Demo ● Results ● Conclusion 10
Overview Overview – Planning with RRT Planning with RRT ● Extend RRT until a nearest vertex is close Extend RRT until a nearest vertex is close enough to the goal state ● Probabilistically complete but converge ● Probabilistically complete, but converge slowly 11
Overview Overview – With Dual RRT With Dual RRT ● Extend RRTs from both initial and goal Extend RRTs from both initial and goal states ● Find path much more quickly ● Find path much more quickly 737 nodes are used 12
Overview Overview – With RRT-Connect With RRT Connect ● Aggressively connect the dual trees using a Aggressively connect the dual trees using a greedy heuristic ● Extend & connect trees alternatively ● Extend & connect trees alternatively 42 nodes are used 13
RRT Connect Algorithm RRT-Connect Algorithm ● Starting from both initial and goal states Starting from both initial and goal states ● Extend a tree and try to connect the new vertex and another tree vertex and another tree ● Alternatively repeat until two trees are actually connect actually connect q goal q init 14
Variations of RRT Connect Variations of RRT-Connect ● Extend & Extend Extend & Extend ● Less aggressive, but works well on nonholonomic constrains nonholonomic constrains ● Connect & Connect ● Stronger greedy ● Stronger greedy 15
Voronoi Region Voronoi Region ● An RRT is biased by large Voronoi regions An RRT is biased by large Voronoi regions to rapidly explore, before uniformly covering the space covering the space 16
RRT Construction Algorithm RRT Construction Algorithm 17
RRT Connect Algorithm RRT Connect Algorithm 18
Results Results ● 0.13s, 1.52s, and 0 13s 1 52s and 1.02s on 270MHz ● I mproves performance by a factor of three or by a factor of three or four in uncluttered environments ● Slightly improves in Slightly improves in very cluttered environments 19
Results Results ● Translations & ● Translations & rotations ● 12s ● 6-DOF ● 4s 20
Conclusions Conclusions ● Reasonably balanced path planning Reasonably balanced path planning between greedy exploration (as in a potential field) and uniform exploration (as potential field) and uniform exploration (as in a probabilistic roadmap) ● Simple and practical method p p ● The huge performance improvements ● The huge performance improvements happen in relatively open spaces only ● Theoretical convergence ratio is not given Theoretical convergence ratio is not given 21
Randomized Kinodynamic Planning Planning Steven LaValle James Kuffner I CRA 1999 # of citation: more than 400 o c tat o o e t a 00 Initial slides from TaeJoon Kim 22
Goal Goal ● Present an efficient randomized path Present an efficient randomized path planning algorithm on the kinodynamic planning problem planning problem 23
Holonomic Path Planning Holonomic Path Planning 24
Nonholonomic Path Planning Nonholonomic Path Planning ● Consider kinematic constraints 25
Kinodynamic Path Planning Kinodynamic Path Planning ● Consider kinematic + dynamic constraints 26
Kinodynamic Path Planning Kinodynamic Path Planning ● Conventional planning: Decouple problems Conventional planning: Decouple problems ● Solve basic path planning ● Find trajectory and controller that satisfies the ● Find trajectory and controller that satisfies the dynamics and follows the path ● [Bobrow et al. 85, Latombe 91, Shiller and Dubowsky 91] [ , , y ] ● PSPACE-hard in general [Reif 79] 27
Outline Outline ● I ntroduction I ntroduction ● Kinodynamic Planning ● Problem Formulation ● Randomized Kinodynamic Planning ● Rapidly-Exploring Random Trees (RRTs) ● Demo ● Results ● Conclusion ● Conclusion 28
State Space Formulation State Space Formulation ● Kinodynamic planning → 2n-dimensional Kinodynamic planning → 2n dimensional state space C denote the C- space X denote the state space x ( q , q ), for q C , x X dq dq dq dq dq dq n 1 2 x [ q q q ] 1 2 n dt dt dt 29
Constraints in State Space Constraints in State Space h h ( ( q q , q q , q q ) ) 0 0 becomes becomes G G ( ( x x , x x ) ) 0 0 , i i for i 1 , ,m and m 2 n ● Constraints can be written in: x f f ( ( x , , u ) ) u U , U : Set of allowable controls or inputs 30
Solution Trajectory Solution Trajectory ● Defined as a time-parameterized Defined as a time parameterized continuous path T : [ [ 0 0 , T ] ] X X , satisfies ti fi the th constraint t i t s free x ● Obtained by integrating Obt i d b i t ti f f ( ( x , u ) ) ● Solution: Finding a control function u : [ 0 , T ] U 31
Randomized Kinodynamic Planning Randomized Kinodynamic Planning ● Randomized potential fields Randomized potential fields ● [Barraquand and Latombe 91, Challou et al. 95] ● Set u which reduces the potential ● Set u which reduces the potential ● Leads oscillations ● Hard to design good potential fields ● Hard to design good potential fields ● Randomized roadmap ● [Amato and Wu 96 Kavraki et al 96] ● [Amato and Wu 96, Kavraki et al. 96] ● Hard to connect two configurations (or states), except for specific environments [Svestka and p p Overmars 95, Reeds and Schepp 90, Bushnell et al. 95…] 32
Rapidly Exploring Random Tree Rapidly-Exploring Random Tree ● A growing tree from initial state A growing tree from initial state 33
Rapidly Exploring Random Tree Rapidly-Exploring Random Tree ● Extend a new vertex in each iteration Extend a new vertex in each iteration u q new q near q q init 34
Results Results – 200MHz, 128MB 200MHz 128MB ● Planar translating ● Planar translating ● X= 4 DOF ● Four different ● Four different controls: up, down, left, right , , g forces ● 500~ 2,500 nodes ● 5~ 15sec ● Planar TR+ RO ● Planar TR+ RO ● X= 6 DOF ● 13,600 nodes 13 600 d ● 4.2min 36
Results Results – 200MHz, 128MB 200MHz 128MB ● 3D translating 3D t l ti ● X= 6 DOF ● 16,300 nodes ● 4.1min ● 3D TR+ RO ● 3D TR+ RO ● X= 12 DOF ● 23,800 nodes 23 800 d ● 8.4min 37
Conclusions Conclusions ● Take advantages from both randomized Take advantages from both randomized potential fields and roadmaps ● “Drives forward” like potential fields ● Drives forward like potential fields ● Quickly and uniformly explores like roadmaps ● Efficient and reliable method ● Efficient and reliable method ● Practical! 38
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